Download Physics 100

CALIFORNIA STATE UNIVERSITY, SAN BERNARDINO
Physics 100
Physics in the Modern World
John McGill
Revised 2007:
Diana Wall and Linh Phan
Revised 2009:
John McGill and James Sheu
Revised 2012: Diana Wall
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Table of Contents
1. Units and Measurements
4
Principles: Units, Measuring, Volume, Density, Uncertainties, Significant Digits
Equipment: Ruler, large beaker, graduated cylinder, scale, density set
2. Variables and the Pendulum
12
Principles: Variables, Graphing, the Pendulum, Hypothesis Testing
Equipment: Meter stick, mass set, lab stand with protractor, string, stop watch
3. Free Fall
23
Principles: Motion: Speed and Acceleration, Gravity
Equipment: Ruler, Computer, Science Workshop interface, Photogate,
Picket fence
4. Levers
30
Principles: Levers, Work, Balance
Equipment: Balance bar and stand, mass set
5. Conservation of Energy
37
Principles: Hooke’s Law, Weight, Energy: Kinetic, Gravitational, Elastic and
Conservation
Equipment: Meter stick, mass set (new), lab stand & attachments, spring, photo
gates and timer
6. Specific Heat
48
Principles: First, Second and Third Laws of Thermodynamics, Specific Heat
Equipment: Thermometer, water heater, calorimeter, lead, copper, and aluminum
objects
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7. Waves
54
Principles: Waves, Wavelengths, Frequency, Standing Waves, Resonance
Equipment: Meter stick, mass set, mechanical wave driver, signal generator,
string, pulley, lab stand with hook, wires
8. Electricity
62
Principles: Electricity, Electric Charge, Electromagnetic Fields, Electric Circuits,
Current, Voltage, Alternating and Direct Currents, Ohm’s Law, Resistance,
Electric Power
Equipment: Analog voltmeter and ammeter, hand held generator, battery pack,
resistor, series resistors, parallel resistors, light bulb and base, wires
9. Electromagnetism
72
Principles: Electromagnets, Magnetic Fields, Induction, Transformers
Equipment: 200, 400, and 800 turn coils, transformer core, signal generator, bar
magnets, battery pack, compass, light bulb and base, galvanometer
10.Light – Reflection and Refraction
82
Principles: Light, Reflection, Refraction, Snell’s Law, Lenses, Mirrors, Optical
Instruments
Equipment: Optical box
11.Appendix A-1
96
Units and Prefixes
12.Appendix A-2
97
Fundamental Constants
13.Appendix B
98
Significant Digits and Uncertainties
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Experiment 1
Units and Measurement
Theory
The discipline of physics is an endeavor to understand the most fundamental principles of
nature. We attempt to explain what can happen in nature in terms of quantitative
relationships between motion, interactions, and states of matter. To do this, we must
assign quantitative values to the properties of matter and describe their relationships
through equations. Thus, physics is highly mathematical. In order to test quantitative
relationships empirically, we must measure the quantities involved. Consequently, the
primary focus of our first experiment is measurement. Each experiment will generate
empirical results which we will compare to the predictions made by established physical
theory.
Units of Measurement
All measurements are made in terms of units. When one reports a measurement it is not
sufficient to just report a number. For instance, it would be meaningless to say that the
width of an object is 5 without specifying whether the width is 5 inches or 5 centimeters
or 5 meters or 5 miles. Each different type of measurement has a type of unit associated
with it. Some of the most basic types of measurements are of length, of time, and of
mass. Each of these has a unit or units of measurements associated with it. The basic
units we will be using for each of these types of measurements are:
Length:
Time:
Mass:
meter (m)
second (s)
gram (g)
Most of the other types of measurements can be expressed in units which are
combinations of these three basic units. For other units of measurement see Appendix A.
Sometimes we will want to work with larger or smaller versions of these basic units. For
instance the mass of a typical person is about 75,000 g. So it might be better to write this
in terms of kilograms (kg), 1 kg = 1000 g. So the mass of a typical person is 75 kg. The
width of your little finger is about 0.01 m or 1 cm (1 meter = 100 centimeter). Another
unit we might find useful is the millimeter (mm), 1 m = 1000 mm, 1 cm = 10 mm.
Metric Prefixes
It can be helpful to know the meaning of some of the prefixes used in the metric system.
These prefixes make it easier to think about really large and really small numbers. You
can find a list of these in Appendix A-1.
There are other basic units that could be used for making measurements such as inches,
feet, miles, etc…, but we will be using only metric units (meters, grams, etc…) in our
class. This is known as the Système International d'Unités or SI Units.
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In addition to measurements of length, mass, and time, we can make other types of
measurements such as area, volume, density, force, energy, and speed. These quantities
can be expressed as combinations of the basic units. For instance, we can measure speed
in miles/hour. The unit is “miles divided by hours”, or miles per hour. What would be
an example of a metric unit of speed? Another example: the area of a rectangular region
whose dimensions are (3 cm) × (2 cm) would be 6 cm2. The unit of area is cm2, which is
just cm × cm.
Volume
In our experiment this week we will be measuring the volumes and masses of some
objects and using those measurements to determine the object’s densities. The volume of
a block is equal to the product of its three dimensions (length × width × height). For
example, the block pictured below has dimensions 2 cm × 2 cm × 3 cm so the volume is
12 cm3. One unit of volume is cm3.
Imagine chopping it up into 12 smaller cubes, each 1 cm on a side. The volumes of each
of the smaller blocks add together to create the volume of the larger block. Objects with
irregular shapes also have volume associated with them. One way of measuring this
volume would be to chop the object up into tiny cubes and add the volumes of each of the
tiny cubes together to find the total volume. Can you think of a way to measure an
irregularly shaped object’s volume without
chopping it up?
2 cm
2 cm
3 cm
One unit of volume is cm3. Some other units
of volume are m3 and mm3. Another very
useful unit of volume is the liter (L). 1 L =
1000 cm3. Consequently, 1 milliliter (mL) = 1
cm3.
Density
Density is another property of an object that can be determined by measurement. The
density of an object or material is defined as the amount of mass per unit volume. So,
one unit of density would be g cm 3 . We can determine the density of an object by
measuring both the mass and the volume of an object and then dividing the mass by the
volume. For example, suppose the 12 cm3 block pictured before has a mass of 24 g.
Then the density would be:
density = mass/volume = 24g/12 cm3 = 2.0 g/ cm3.
The density in g/ cm3 represents the mass of each 1 cm3 chunk of the block (assuming the
density is uniform).
Density is often a property of the material and does not depend on how much material
you have. For instance, lead has a density of 11 g cm 3 and water has a density of 1.00
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g cm 3 . The mass of one cubic centimeter of water was the original definition of the
gram.
Measuring Instruments
Different kids of measuring instruments are used to measure different kinds of quantities.
For instance, one might use a meter stick to measure length, a stop watch to measure
time, a scale to measure mass, and a thermometer to measure temperature. Different
kinds of measuring instruments are used to measure quantities of differing sizes. It
would be impossible to measure the size of a bacterium with a meter stick. You couldn’t
use the same instrument to weigh a truck as you do to weigh an ounce of water. What
kinds of instrument could be used to measure distances of miles?
Measurements and Uncertainties
No measurement can ever be exact. The accuracy of a measurement (that is to say how
close it is to the truth) will generally depend on the measuring instrument and the care
taken by the measurer. Every measuring instrument has a limit on how precisely it can
be read. For instance, a device with a digital display (like a digital watch) has a finite
number of digits, limiting its precision. No finer gradation can be determined than the
size of the least significant digit. Another example would be a meter stick which has a
smallest division of 1 mm, or 0.1 cm.
When we record a measurement we indicate our uncertainty in two ways. First, we only
write down digits until we reach a digit we are unsure about. In other words, we should
be sure about all the digits except the last one. Second, we indicate a range of possible
values with a ± and an uncertainty estimate. This estimate is a judgment we make based
on the measuring instrument and our experience. Based on my experience and the fact
that the smallest division on a meter stick is 1 mm = 0.1 cm, I believe that any
measurement I make with a meter stick is uncertain by at least ±0.1 cm.
Other measuring instruments can give more precise, thus more accurate, measurements.
A meter stick is limited by the size of its smallest division (0.1 cm). Vernier calipers, on
the other hand, can be used to make much more precise measurements. However, with
most increases in precision comes a limitation. In the case of the calipers their maximum
range of measurement is about 20 cm.
Uncertainties in calculated results
Since the actual value of a measurement is uncertain, any result calculated from that
measurement is uncertain. For example, suppose we measured the three dimensions of a
wood block to be 8.3±0.1cm x 10.6±0.1cm x 5.2±0.1cm. That means that the volume of
the block will be uncertain as well.
Significant Figures or Digits
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We can use the concept of significant figures (or digits) to determine the approximate
uncertainty of a calculated result like the volume. Every time you record a measurement,
that measurement is recorded to a number of significant digits. Remember that you only
record the measurement up to the first digit you are uncertain about. Often the number of
significant digits will be the same as the total number of digits. This is true of our
measurements of the wood block above. However this is not always the case.
For rules on significant figures and rounding, as well as more on uncertainties, see
Appendix B.
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Your Name: ____________________
Lab Partner: ____________________
Experiment 1
Work Sheet
I. Mass
Measure the mass of each of the seven objects listed in the following table.
Record each measurement with an uncertainty estimate.
Blocks
Object
Aluminum
Brass
Cylinders
Wood
Aluminum
Brass
Plastic
300 mL water
Mass (g)
Discussion: How do we measure the mass of 300 mL of water?
II. Measuring Dimensions
Measure the three dimensions of each of the three blocks with the meter stick and record
each result in the following table.
Meter Stick Measurements
length
Blocks
Aluminum
Brass
Wood
width
Length (cm)
height
± 0.1cm
Width (cm)
± 0.1cm
Height (cm)
± 0.1cm
3
Volume (cm )
Discussion: What is the equation for the volume of a block?
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Cylinders
Aluminum
Brass
Plastic
diameter
Diameter (cm)
± 0.1cm
height
Height (cm)
± 0.1cm
3
Volume (cm )
Discussion: What is the equation for the volume of a cylinder?
III. Measuring Volume Directly
Discussion: How can we measure volume directly?
Measure the volumes of the three cylinders and the Aluminum and Brass blocks directly
(without using the ruler measurements). Fill in the uncertainty in your measurement.
Direct Measurement of Volumes
Aluminum
Brass
Plastic
Block
± ____mL
Cylinder
± ____mL
Disscussion: Are the volumes measured directly exactly the same as those obtained from
the dimension measurements (part II)? If not, are the values obtained by the two
methods consistent given the margin of error in the direct measurements?
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IV. Density
Compute the density of each of the objects from the measurements you have made.
Remember: density = (mass/ volume ). Use the value for the volume obtained in part II.
Record the result in the following table. Use your data to find the density of water and
record the result in the density table. This is the experimental value; do not just use the
theoretical value of 1.00 g/cm3. Remember significant figures. Label each object F or S
according to whether the object floats or sinks.
3
Object
Density (g/cm )
Float/Sink
Block
Aluminum
Cylinder
Block
Brass
Cylinder
Water
Plastic
Cylinder
Wood
Block
Discussion: Do you see a relationship between density and buoyancy (the tendency to
float)?
Discussion: Look at the density values of your brass block and brass cylinder, are the two
values exactly the same? Are the two values for the aluminum block and aluminum
cylinder exactly the same? Should they be? What are the possible reasons that the two
values do not agree?
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V. Conclusion
Discussion: Ships are made of steel, which is much denser than water. How can steel
ships float?
Write a short paragraph explaining what you learned in this experiment.
What were the things you did not understand in this experiment?
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Experiment 2
Variables, the Pendulum and Oscillations
Theory
Variables
Often in physics we are interested how changing some aspect of a system changes its
behavior. For instance, one might ask how changing the height from which an object
falls changes the time it takes it to fall to the ground. What would you expect this
relationship to be? As always, we are interested in the quantitative as well as the
qualitative aspects of the problem. Any quantitative property of a system that can change
is called a variable. In the case of the falling object, there are two variables of interest to
us, the height from which it falls and the time it takes to reach the ground. In general
there are two different types of variables that we can have in an experiment. There are
variables that we have direct control over and whose values can be chosen before the
experiment is begun. These are called independent variables. In the case of our falling
object experiment, the height is an independent variable. The second type of variable is a
variable whose value we do not know before doing the experiment. Its value depends on
our choice of independent variables. This second type is called a dependent variable. In
the falling object experiment, the time it takes the object to fall would be a dependent
variable. In the falling object experiment, we could drop the same object from several
different heights, and measure the time it takes to fall from each height with a stop watch.
We could choose the heights we wanted to use before performing the experiment. Let’s
say we chose 5m, 10m, 15m, 20m, 25m, and 30m. If we perform the experiment we
might get results like those shown in the table.
Height (m)
Drop Time (s)
5.0
10.0
15.0
20.0
25.0
30.0
1.0
1.4
1.7
2.0
2.2
2.4
± 0.1m
± 0.1s
As you can see the drop time increase as the height
increases. Is this consistent with our expectations?
Graphs
It is useful to display the relationship between two variables in a form of a graph. A
graph is an x-y plot of the data points. Usually, the independent variable is plotted on the
horizontal or x-axis and the dependent variable on the vertical or y-axis. When we make
a graph, we choose the scale of the graph axis such that the data fill up most of the graph.
Below is a graph of the falling object experiment data.
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I
Drop Time versus Height
3
2.5
Drop Time (s)
2
1.5
1
0.5
0
5
10
15
20
25
30
Height (m)
35
The symbol , represents the range
of uncertainty in each data point.
These are called error bars. Notice
that each axis is clearly marked
with the name of the variable, the
unit, and the scale at regular
intervals on the grid. These are all
important elements that should be
present on a graph. It is also
worthy of note that the axis doesn’t
always have to start at zero. If you
look at the drop time axis, you’ll
see that it starts at 0.5s not at 0s.
Sometimes it is useful to graph some calculated result of the data. For instance you
notice that the data points on the first graph don’t fall near a straight line, but if instead
we graph the square of the drop time versus the height we get:
As you can see, the data points on
the drop time squared versus height
graph fall much closer to a straight
line than those on the drop time
versus height graph.
Drop Time Squared versus Height
7
Drop Time squared (s2)
6
5
Proportionality
4
3
When the graph of two variables
falls on (or very near, given
2
uncertainties and errors inherent in
1
any experiment) a straight line, we
say that there is a linear relationship
0
15
20
25
30
35
0
5
10
between the two variables graphed.
Height (m)
When the straight line can be fit to
the data being graphed, we say that they are proportional to each other. So we would say
that the Drop Time Squared is proportional to the Height from which the object is
dropped since the data fall on (or very near) a straight line. When two variables are
proportional to each other the ratio between them is constant. This is apparent for the
Drop Time Squared versus Height data as shown in the following table.
2
2
2
2
Drop Time Squared D (s )
Height H (m) ± 0.1m
± 0.02 s 2 / m
Ratio D /H (s /m)
5
0.20
1
± 0.1m
10
0.20
2.0
± 0.2m
15
0.19
2.9
± 0.3m
20
0.20
4
± 0.4m
25
0.19
4.8
± 0.5m
30
0.19
5.8
± 0.6m
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As you can see from the table, the ratio of the Drop Time Squared (D2) divided by Height
(H) is constant, within the allowed uncertainty. The ratio of two variables that are
proportional is called the constant of proportionality.
Slopes
Drop Time Squared versus Height
2
Drop Time Squared (s2)
(s )
7
6
5
4
3
2
1
0
0
10
20
30
Height (m )
In other words: slope =
40
On a graph, this ratio is
known as the slope of the
line. To determine the
constant of proportionality
using a graph that you drew
using graph paper, select
two points on the line (these
two points may not be data
points since data points
usually are not on the line
due to experimental errors)
where point 1 is (x 1 , y1 )
and point 2 is (x 2 , y2 ).
Select points that intersect
on the graph paper. The
slope is defined as the
change in y over the change
in x (the rise over the run).
( y 2 − y1 )
(x2 − x1 )
Sometimes (0,0) is a valid first point but for example’s sake, we will not use this point.
The two points we will select are those that intersect on the graph paper and are circled.
The values of the points can be read off the scales on the axis.
Point 1 (2m, 0.5s2)
Point 2 (18m, 3.5s2)
(3.5s
slope =
)
− 0.5s 2
3.0 s 2
=
= 0.19 s 2 / m
(18m − 2m ) 16m
2
Notice that the constant of proportionality found in the graph’s slope matches that from
the data table when the ratios were calculated individually. Graphs are preferred since
they can forecast trends beyond the data we have collected. They are also preferred
because they are a pictorial representation of the data and they make noticing trends
easier.
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The Pendulum
A pendulum consists of a bob of
mass which is allowed to swing
back and forth on the end of
string (or other long thin arm
which is attached to a fixed point.
A
L
The pendulum can be
characterized by several
variables: the mass (m) of the
bob, the length (L) of the arm,
m
the amplitude (A) or the swing
angle, and the oscillation period
(T). The oscillation period is
defined as the amount of time it
takes for the pendulum to swing
from one extreme to the other and back again (one cycle). Notice that a single letter is
chosen to represent each variable. These letters are useful as shorthand for referring to
variables especially when writing down equations that relate them to each other. In an
experiment involving the pendulum, which of these variables are independent? Which
are dependent?
Hypothesis Testing
We often have an idea of what the outcome of an experiment will be before we do it. In
fact, most experiments are designed with the goal of testing a particular theory about how
nature behaves. A belief about what the result of an experiment will be, before the
experiment is done, is called a hypothesis. Hypotheses can be based on common sense,
what we have learned from others or from other experiments, or on some kind of
intuition. When we have a hypothesis about how something will behave we need to
design an experiment to test this hypothesis. The experiment we design should test the
hypothesis as directly as possible, and from the results of the experiment we should be
able to confirm whether what we believe was true or false. Sometimes, the experiment
confirms that we were exactly right. Other times, our hypothesis is shown to be
completely wrong. For example, Aristotle believed that heavier things fall faster than
lighter ones. Galileo showed that, in general, all things fall at the same rate under Earth’s
gravity. In other instances, our experiments show us that our hypothesis, though not
completely wrong, need adjustment.
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Your Name: ____________________
Lab Partner: ____________________
Experiment 2
Work Sheet
I. Pendulum Hypothesis
Today, you are going to make three hypotheses about the results of this experiment. It is
best to make a hypothesis as simple as possible. Following are three questions about how
the oscillation period (time for one cycle) of the pendulum changes when the different
variables are changed systematically. In each case, you decide whether you think that the
oscillation period will increase, decrease, or remain the same (meaning the variable does
not affect the outcome). It won’t matter whether your hypothesis comes out right or
wrong, so just circle your best guess.
1. As we increase the amplitude (leaving the length of the arm and mass of the bob
the same) how will the oscillation period change?
(increase, decrease, remain the same)
2. As we increase the mass of the pendulum bob (leaving the length of the arm and
amplitude the same) how will the oscillation period change?
(increase, decrease, remain the same)
3. As we increase the length of the arm (leaving the mass of the bob and amplitude
the same) how will the oscillation period change?
(increase, decrease, remain the same)
In this experiment, you will test each of these hypotheses.
II. Oscillation Period and Amplitude
protractor
Attach the metal bob to the lab stand. Make the length of
the pendulum about 25 cm. Thread the string through one of
the holes on top and wrap (do not tie) around base of screw.
Tighten the screw to hold the string in place.
Start the pendulum swing with a small amplitude (read the
angle off the protractor) and use the stop watch to measure
how long it takes for the pendulum to go through ten cycles.
ruler
Discussion: Why measure for ten cycles and not just one?
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Do three trials to reduce random uncertainties in the time. Record your values in the
table below. Divide each trial by 10 to obtain the period of one oscillation. Find the
average time for one oscillation of the three trials. To determine the margin of error ( ± ),
subtract the smallest trial from the largest and divide by two. Record your results in the
table.
Small Amplitude: A S = __________
Time for 10 oscillations (s)
Time for one oscillation (s)
Trial 1
Trial 2
Trial 3
Average time for one period (T S ):
±
Repeat the previous procedure with a larger amplitude oscillation. (about twice as large).
Record your results in the table
Large Amplitude: A L = __________
Time for 10 oscillations (s)
Time for one oscillation (s)
Trial 1
Trial 2
Trial 3
Average time for one period (T L ):
±
Does the average time for one oscillation change when you increase the amplitude? If so,
is the change larger than the margins of error allow for? Considering this, does your data
confirm your hypothesis? Explain.
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III. Oscillation Period and Mass
Measure the mass of the lead bob. Suspend the lead bob with a string length of 25cm and
allow it to oscillate with amplitude of 20o. Measure the time for 10 oscillations and enter
it into the table. Repeat this twice for each bob. To find the margin of error subtract the
smallest trial from the largest and divide by two.
Mass (g)
Time 1 (s)
Time 2 (s)
Avg time for 10
oscillations (s)
Lead
±
Brass
±
Steel
±
Wood
±
Does the time for 10 oscillations change as you change the mass? If so, is there a
consistent trend? Considering the margins of error is there a significant difference
between the times? Do you confirm the hypothesis you made about how the period
would change when the mass increases?
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IV. Oscillation Period and Length
Suspend two steel bobs from the lab stand, one with half the length of the other and pull
both with the same amplitude. The only difference between these two bobs is the length
of the arm. Set both pendulums to swing. Which has the longest period of oscillation?
Look back at your hypothesis about the changes in length. Do your observations confirm
your hypothesis? Explain.
Now we will make a methodical series of measurements describing how the oscillation
period changes when we change the length of the pendulum arm. As before, we will
measure the time for ten oscillations but this time we’ll vary the length of the pendulum
each time. Use the steel bob and record the results in the following table. We will only
perform one trial at each length. Use an amplitude of 20o for each measurement.
Length (cm)
Ten periods (s)
Period (s)
Period2 (s2)
20
30
40
50
60
70
80
Discussion: Which variable is the independent variable? Which axis should it be on?
Discussion: Which variable is the dependent variable? Which axis should it be on?
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* Graph the Period versus Length (T vs L).
Make sure you have each of these parts of the graph:
 Title for graph
 Sensible scale for the axis (period scale of about 0 to 2.0 s and length scale from 0
to 80cm)
 Variable name on x and y axis with proper units
 Data points with error bars
 Draw the best fit curve (if it is a linear line, remember to use a ruler)
Each student should attach a graph to this work sheet when it is turned in.
* Graph the Period Squared versus the Length (T2 vs L).
The length scale will remain the same but it may help to change the scale of the period
squared scale (about 0 to 3.0 s2). Make sure you have each part of the graph excluding
error bars for the period squared since this was not determined.
Attach your graph to this work sheet when you turned it in.
Discussion: Which graph fits a straight line? Find the slope of this line. Show your
work on the graph itself.
Slope of the line (experimental): ___________________
From our knowledge of how the simple pendulum is expected to behave, we can predict
what the value of the slope. According to the simple pendulum equation:
L
T = 2π
g
The variables can be rearranged to solve for the period squared (T2) with all the constants
grouped together:
 4π 2 
 L
T 2 = 
 g 
This equation follows the equation for a straight line:
y = mx + b
where the slope is m, the y-intercept is b, y is the y-axis value and x is the x-axis value.
Notice that the y-axis value is T2, the x-axis value is L, and the y-intercept is zero so the
slope must equal the constants (4π2/g). Calculate this constant where g=981cm/s2 and
π≈3.14.
Slope of the line (theoretical): _______________________
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To determine if your values agree it is useful to do a percent discrepancy (%D)
calculation which determines how “discrepant” your value is. In general your percent
discrepancy should be below 10% (in other words you are 90% correct and 10% in error).
Calculate the percent discrepancy between the theoretical value of the slope and the
experimental value of the slope.
%D =
theoreticalvalue − experimentalvalue
× 100%
theoretical value
Before you performed this experiment, you formed several hypotheses about pendulum
behavior. Now, you should have either confirmed or changed these ideas according to
the results of your experiment. Write down what you now believe about the behavior of
the pendulum.
1. As we increase the amplitude (leaving the length of the arm and mass of the bob
the same) how will the oscillation period change?
(increase, decrease, remain the same)
2. As we increase the mass of the pendulum bob (leaving the length of the arm and
amplitude the same) how will the oscillation period change?
(increase, decrease, remain the same)
3. As we increase the length of the arm (leaving the mass of the bob and amplitude
the same) how will the oscillation period change?
(increase, decrease, remain the same)
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V. Conclusion
Write a short paragraph explaining what you learned in the lab.
What were the things you did not understand in this experiment?
- 22 -
Experiment 3 Uniformly Accelerated Motion
Theory
Speed and Acceleration
When an object falls under the influence of gravity, it does not fall at a constant speed. The
falling object’s speed will continually increase as it falls toward the ground. An increase in
speed over time is called acceleration. To further understand speed and acceleration let us
look at a hypothetical experiment given in the previous lab. In this experiment, an object is
dropped from a given height and the amount of time it takes to fall is measured with a stop
watch. The data from this experiment is recorded in the following table.
Height (m)
Drop Time (s) ±
± 0.1m
0.1s
5.0
10.0
15.0
20.0
25.0
30.0
1.0
1.4
1.7
2.0
2.2
2.4
As you might have expected, the higher the object
when it is dropped, the longer it takes to reach the
ground. There are some other features of the fall of
the object under gravity that are worth noting. One
way we can discover Figure 1 shows a graph of this
data. new things about this experiment is by graphing
the data.
Notice how the line drawn through the data points
curves upward. This means the object is accelerating as it falls. When an object is
accelerating it moves faster and faster as it goes. What happens to your car as you get on the
freeway is another example of acceleration. When you start out at the bottom of the entrance
ramp to the freeway, you might be starting from as little as 0 miles/hour, but when you merge
with the traffic on the freeway, you are traveling at freeway speeds (60 – 75 miles/hour). So
somewhere in between your speed must have been increasing.
- 23 -
Another way to
represent the motion
of the falling object is
to graph the speed
of the object as it falls
to the ground. Figure
2 is a graph of the
speed of the falling
object versus time.
You can see from the
graph that the speed
really does increase
with time.
Figure 2
Another thing that you might notice is that the line drawn through the data points is straight.
This means that the acceleration is constant. All objects falling under the influence of
gravity, near the surface of the earth fall with constant acceleration. In fact, the acceleration
of all objects falling under gravity is the same no matter what the size or shape of the object.
The air drag force exerted on lighter objects like feathers and sheets of paper does cause
them to fall more slowly, but for larger more dense objects like books and bricks and people
this effect is negligible at low speeds. Figure 3 is a graph of the acceleration of the falling
object versus time. Note that the acceleration does not change over time. Also notice that the
2
acceleration is about 5m/s . Look at Figure 2 and notice that the ball’s speed increases by
5m/s every second.
Acceleration (m/s 2)
Accelertion of a dropped ball over time.
10
8
6
4
2
0
0.0
0.5
1.0
1.5
2.0
Time (s)
Figure 3
- 24 -
2.5
As you might have noticed,
speed and acceleration are
measurable quantities, and in
the course of this experiment
you will have to measure
them. You are probably
already familiar with
measurement of speed. As
you drive your car you are
constantly measuring your
speed by looking at the
speedometer on your dash
board.
In this experiment, we will be measuring speed in a more basic fashion. The speed of an
object is the distance it travels (along a straight line) divided by the amount of time it takes to
travel that time.
For example, if you travel a distance of 120 meters over a period of four seconds (starting
from rest):
d f − di
s ave =
t f − ti
(
(
)
)
s ave = average speed
t f = final time
d f = final distance
d i = initial distance
t i = initial time
(120m − 0m ) = 120m = 30 m s
s ave =
(4s − 0s )
4s
The unit for speed is meters per second (m/s). Notice that the average speed was 30 m/s.
The instantaneous speed could have fluctuated up or down a great deal over that time period,
but on average it was 30 m/s.
The quantification of acceleration may be new to you, but acceleration is a measurable
quantity just like speed. Acceleration is the change in the speed per unit time. In other
words, the acceleration of an object is equal to the amount that the speed changes divided by
the amount of time it takes it to change. The idea of average applies to the quantity of
acceleration just like it does to speed. You can calculate acceleration just like you can
calculate speed.
For example, you accelerate in your car as you are getting on the freeway. You start from a
speed of 0 m/s and accelerate up to 30 m/s (about 60 miles/hour), and you do this over a eriod
of 15 seconds. Then your average acceleration is:
a=
(s
(t
f
− si )
− ti )
a = acceleration
s f = final speed
t f = final time
f
si = initial speed
a=
(s
(t
f
f
− si )
t i = initial time
(30m / s − 0m / s ) = 30m / s = 2m / s 2
(15s − 0s )
15s
− ti )
=
The unit of acceleration is meter per second squared (m/s2). In the example, the car increases
its speed by 2 meters per second, every second.
- 25 -
Your name ____________________
Lab Partner ___________________
Experiment 3
WORK SHEET
PROCEDURE
In this activity, you will drop a “picket fence” (a clear plastic strip with uniformly spaced
opaque bands) through a photogate. The photogate beam is blocked by each opaque band
and the time from one blockage to the next becomes increasingly shorter. Knowing the
distance between the leading edge of each opaque band, the Science Workshop program
calculates the average speed of the picket fence from one band to the next. A graph of
average speeds versus time can give the acceleration due to gravity of the falling object.
PART I: Computer Setup
1.
Connect the Science Workshop interface to the computer, turn on the interface and
then turn on the computer.
2.
Connect the photogate’s stereo phone plug to Digital Channel 1 on the interface.
3.
Open the Science Workshop file titled “ Free Fall”.
•
The document will open with a Graph display that has plots of Position versus Time
and a Table of Position versus Time.
PART II: Equipment Setup
•
The Science Workshop program has a 5.0 cm (0.050 m) spacing, leading-edge-toleading-edge, for the opaque bands on the picket fence.
1.
Turn the photogate head of the accessory photogate sideways so that you can drop a
picket fence vertically from above the photogate and have the picket fence move
through the photogate’s opening without hitting the photogate.
Freely Falling Picket Fence
Picket fence
Photogate
To Interface
- 26 -
Base and
support rod
Part III: DATA Recording
1.
Prepare to drop the picket fence through the photogate beam. Hold the picket fence
at one end between your thumb and forefinger so the bottom edge of the picket
fence is just above the photogate beam.
2.
Click the “START” button and then drop the picket fence through the photogate
beam. Remember, data collection begins when the photogate beam is first blocked.
3.
When the picket fence is through the beam, click “STOP” to end recording.
4.
Record the value for the Time in the Data Table.
5.
Calculate the average speed column by dividing the change in the distance (5.0 cm)
by the time interval. The time interval is the difference between successive times.
6.
Calculate the time at average speed column by averaging the two successive times
together.
7.
Make a graph from the Data Table, Average speed versus Time at average speed.
Time (s)
Distance (cm)
Time at average speed (s)
5.0
10.0
15.0
20.0
25.0
30.0
slope of velocity versus time = ______ (from Graph)
acceleration = ______ (theoretical value)
- 27 -
Average Speed (cm/s)
IV. Questions
1.
How does the slope of your velocity versus time plot compare to the accepted value
of the acceleration of a free falling object (g = 980 cm/s2)?
% difference =
2.
Theoretical value - exp erimental value
x100%
Theoretical value
What factors do you think may cause the experimental value to be different from
the accepted value?
- 28 -
V. Conclusion
Write a short paragraph explaining what you learned in this experiment.
What were the things you did not understand in this experiment?
- 29 -
Experiment 4 Levers
Theory
Levers
Levers are simple machines that allow you to lift or move something you otherwise would
not be able to move. You have all heard the phrase “apply leverage” in many different
contexts, and most of you have used a lever of some type or another. When you use a
wrench to tighten or loosen a bolt, you are using a lever. When you use a wheelbarrow to
carry or dump a heavy load, you are using a lever, and when you use a crow bar to pry open a
crate, you are using a lever. In each of these cases the lever allows you to lift or move
something you would not be able to lift or move with your bare hands or fingers. This can
happen because a lever allows you to exert more force than you could by simply pushing,
pulling or lifting. Pictured below is a common type of lever.
Force applied to end
of lever (F2)
In general a lever consists of a long rigid bar, and a fulcrum. The fulcrum is the point about
which the bar pivots. The load is whatever is to be moved. The load arm is the distance from
the fulcrum to the load, and the lever arm is the distance from the fulcrum to where the force
(push, pull or lift) is applied. The lever above could be used to lift a mass that was too heavy
to be lifted with arms and legs alone. Sometimes the load and the applied force are on the
same side of the fulcrum, as pictured below. This is true in the case of the stapler and wheel
barrow.
In either case, the leverage or mechanical advantage that you gain by applying a lever
increases as the lever arm increases and decreases as the load arm increases. So, in general,
- 30 -
you want the lever arm to be as large as possible and the load arm to be as short as possible.
Work
At first it may seem that the lever violates Newton’s Third law. This states that for every
action there is an equal and opposite reaction. How can the force exerted on the load be
larger than the force exerted by the person? We can reconcile the lever to Newton’s laws by
realizing that there are forces other than the force exerted at the lever arm at work. The
fulcrum also exerts a force on the bar which is transferred to the load. For this reason it is
necessary that the fulcrum be firmly secured to the earth or some other very massive object.
If the fulcrum is not firmly secured the lever may not work.
As students of physics we are not only interested in the qualitative aspects of how the lever
works. We are also interested in the quantitative aspects of the lever. One might ask how
much mass can be lifted with a given lever arm and load arm. The best way to describe this
quantitatively is to use the concept of Work. The amount of work done by an applied force
is equal to the force applied times the distance moved. For example if one lifts a 1 kg mass
to a height of 1 m the work against gravity is the product of the gravitational force and the
distance moved.
2
2
2
Work = Force x Distance = 9.81 m/s x 1 kg x 1 m = 9.81 kg m /s = 9.81 Joules
In the case of the lever the work
done on the load is equal to the
work done by the applied force
at the lever arm. For example if
2m
the load arm is 1 m long and
the lever arm is 2 m long and the
load has a mass of 1 kg, and the
load is lifted one meter, then the
amount of work done on the load is 9.81 Joules. However, when the load arm moves 1 m the
lever arm moves two meters, so the force applied at the lever arm must be only half of that
2
2
applied to the load. The force applied to the lever arm is (1/2)(9.81 kg m/s ) = 4.91 kg m/s .
2
The unit of force kg m/s is called a Newton and abbreviated N. In general, the work done on
the load is equal to the work done by the applied force at the lever arm. Since the work done
by a force is equal to the force multiplied by the distance moved:
W1 = W2 F1 D1 = F2D2
- 31 -
Consequently,
F1/ F2 = D2/ D1
The lever and load
arms and the
vertical movements
are geometrically
proportional. So
the ratio of the load force to the lever arm force is equal to the inverse ratio of the arms’
radii.
F1/ F2 = r2/ r1
Balance
The relation above is the basis for the balance. You all know that if you suspend equal
weights at equal distances from the fulcrum the result will be balance. Neither mass will rise
nor fall. If you move one of them farther out on the rod it will begin to fall and the other will
rise.
One thing that you may not realize is that you can balance a heavy object with a lighter object
if the lighter object is farther from the fulcrum.
This is a direct consequence of the relationship of the force to arm length derived in the
previous section. The force of gravity pulls down on each of the masses. The first mass, m1,
at length r1 from the fulcrum experiences a gravitational force of Fg1 = m1g. The second
mass, m2, experiences a gravitational force
Fg2 = m2g. If the masses are balanced then
the masses are at rest, which means that the
force of gravity is being counteracted by the
force exerted on each mass by the rod. We
know from the earlier discussion that the
force exerted on m1 by the rod, F1, is related
to the force exerted on m2 by the rod, F2.
𝐹1 𝑟2
=
𝐹2 𝑟1
- 32 -
We also know that F 1 and F 2 are equal and opposite to the gravitational force on the
masses.
𝐹1 = 𝐹𝑔1 = 𝑚1 𝑔
𝐹2 = 𝐹𝑔2 = 𝑚2 𝑔
Consequently, there is a simple relationship between the masses and the distances to the
fulcrum.
𝑚1 𝑟1 = 𝑚2 𝑟2
- 33 -
Your Name: ____________________
Lab Partner: ____________________
Experiment 4 Work Sheet
I. Balance
Remove the two mass holders from
the bar. Use the scale to measure the
mass of each of the two hanging
masses along with its corresponding
mass holder. Note that the value
will be larger than what is printed on
the hanging mass because of the
weight of the mass holder.
Small Mass: m 1 = _____________ Large Mass: m 2 = _____________
Then find the ratio of the two masses, m2/m1. Record your result as a decimal number.
𝑚2
𝑚1
= _____________
Place the smaller mass, m 1 , at r 1 =10 cm from the fulcrum (0 cm) and find the point r 2 on the
other side where the larger mass, m 2 , balances it. Repeat this procedure with the smaller mass
at r 1 = 12, 14, 16, 18, and 20 cm from the fulcrum. Make sure that the middle of the bar (0
cm) is always at the fulcrum. Record your results in the following table. Find the ratio of
for each pair of measurements and record the result in the third column of the table as a
decimal number.
r1 (cm)
10.0
12.0
14.0
16.0
18.0
20.0
r2 (cm)
𝑟1
𝑟2
Table 1 for balance points
𝑟1
𝑟2
Discussion: What is the relationship
between
𝑚2
𝑚1
show this?
- 34 -
and
𝑟1
𝑟2
?
Does your data
Repeat the above procedure with a different pair of (unequal) masses.
Small Mass: 𝑚1 = _________________________
Large Mass: 𝑚2 = ________________________
Then find the ratio of the two masses.
𝑚2
= _______________________
𝑚1
Discussion: What do you expect the value of
𝑟1
𝑟2
to be?
Table 2 for balance points
r1 (cm)
𝑟1
𝑟2
r2 (cm)
10.0
12.0
14.0
16.0
18.0
20.0
Discussion: Compare the values of
results show that
𝑚2
𝑚1
=
𝑟1
𝑟2
?
𝑟1
𝑟2
recorded in the table to the value of
- 35 -
𝑚2
𝑚1
. Do your
II. Conclusion
Write a short paragraph explaining what you learned in this experiment.
What were the things you did not understand in this experiment?
- 36 -
Experiment 5
Conservation of Energy
Theory
Conservation of Energy
This experiment will demonstrate the law of conservation of energy. The law of
conservation of energy states that energy can neither be created nor destroyed. So in an
isolated system (a system is a collection of bodies, an isolated system is a collection of
bodies that is not acted on by anything external to the system) the total amount of energy
never changes. Even though the total amount of energy never changes, energy can be
transferred from one body to another, and transformed from one form to another. There
are many forms of energy: kinetic energy, gravitational energy, thermal energy, elastic
energy, chemical energy, electrical energy, and nuclear energy.
In this experiment, you will only deal with kinetic energy, gravitational energy and
elastic energy. There will be other forms of energy in the system of bodies you will be
working with, but those forms of energy will not be changing in significant amounts to be
considered.
Kinetic Energy
Kinetic energy is the energy of motion. Any object that is in motion has kinetic energy.
In Newtonian physics, the amount of kinetic energy, K that a body has is given by:
KinE = (1/2) × mass × (speed squared) = (1/2)mv2
where m is mass and v is speed. Notice that the kinetic energy is proportional to the mass
and proportional to the speed squared.
m
speed = v
KinE = (1/2)mv2
Units of Energy
You will be measuring masses in kilograms (kg), lengths in centimeters (m) and time in
seconds (s). So, for example, a mass of 10.0 kg moving at a speed of 4.0 m/s has a
kinetic energy of:
KinE = (1/2)mv2 = (1/2) (10.0kg)(4m/s)2 = 80 kg m2/s2 = 80 Joule = 80 J
The unit, kg m2/s2, is called a Joule (J). So the Kinetic energy is K = 80 J.
Another unit of energy is the erg. 1 erg = 1 g cm2/s2
- 37 -
Gravitational Energy
m
h
GravE= mgh
Gravitational energy (often called gravitational potential
energy) is the energy associated with gravity.
Gravitational energy is associated with the height of an
object. Notice that Gravitational energy is proportional to
the mass and proportional to the height.
The amount of gravitational energy, GravE, that a body
has (actually, we think of the energy being shared
between the Earth and the body) is given by:
Gravitational energy = weight of the body × the height of the body
Remember: Weight = mass × 9.8 m/s2 = mg
GravE= mgh
where m is the mass of the body, g is gravitational acceleration (9.8 m/s2) and h is the
height.
Elastic Energy
Elastic energy (often called elastic potential energy) is the energy associated with
stretching or compressing a spring (or stretching or compressing anything that can be
stretched or compressed). The amount of elastic energy in the spring depends on how
much the spring is stretched or compressed.
Hooke’s Law
Most springs obey Hooke’s Law, which means that the force required to stretch the
spring is proportional to amount that it is stretched. For a spring that obeys Hooke’s Law
the force, F, that must be exerted on the spring to make it stretch a distance x is given by:
- 38 -
Force exerted on spring = (spring constant) × (amount of stretch)
F = kx
where k is the spring constant, and x is the amount of stretch.
We will use lower case k for the spring constant and upper case K for kinetic energy.
The amount of stretch, x, is the equal to the length of the spring when it is stretched
minus the length of the spring when it is not stretched.
For a spring that obeys Hooke’s Law, the amount of elastic energy, ElastE, stored in the
spring is:
Elastic energy = (1/2) × (spring constant) × (amount of stretch squared)
ElastE = (1/2) kx2
Mechanical Energy
You will be measuring the kinetic energy, the gravitational energy, and the elastic energy
of a mass oscillating up and down on a spring. There will be other forms of energy in
your experiment that you will not measure (i.e. Thermal energy). However, these other
forms of energy will not change substantially during your experiment. You will treat the
sum of the kinetic, gravitational and elastic energies as if it were the total energy in the
system. We will call the sum of the kinetic, gravitational and elastic energies the total
mechanical energy:
Total Mechanical Energy = Kinetic Energy + Gravitational Energy + Elastic Energy
E total = KinE +GravE + ElastE
You will measure the total mechanical energy at three points in the oscillation of the
mass on the spring, at the top, in the middle and at the bottom. What you will find if you
do the experiment carefully is that the total mechanical energy doesn’t change
appreciably throughout one oscillation of the mass on the spring.
- 39 -
Your Name ______________________
Lab Partner ______________________
Experiment 5
Work Sheet
I. Measurements
1. Middle Height: Allow the 0.500 kg mass to hang freely
from the spring. Stop it from oscillating (bouncing) so that
it hangs at rest. While its hanging at rest measure the height
of the bottom of the mass above the table. Measure all
heights and lengths in meters. This is the middle height,
HM. Record your measurement below: (you will need to
convert all length measurements to meters)
H M = _________________________m
2. Position the photogate so that mass hangs in the middle
of the infrared beam both vertically and horizontally.
Middle
Height HM
4. Timer Setup: Make sure that the timer is plugged in and the power strip is on.The
timer setting should be as follows. The timer should be set to GATE. The memory
switch should be on. The precision switch should be set to 0.1 ms. As shown in the
photo below. Use the red reset button to set the timer to zero before every measurement.
0.1 ms
Gate
Reset to
zero
Memory - on
- 40 -
4. Top Height: Lift the 0.500 kg mass by the stem until
the spring is unstreched. Hold the mass in the position
where it barely touches the spring but does not stretch it.
Measure the height of the bottom of the mass above the
table. This is the top height, H T . Record your
measurement below:
H T = _________________________m
Top
Height HT
5. While holding the mass by the stem so that the spring
is unstretched. place the meter stick near the mass so
that you will be able measure the bottom most height
that the 0.500 kg mass reaches while it is oscillating.
Reset the time to zero by pushing the red reset button.
6. Drop the 0.500 kg mass. Within first two or three
bounces measure the lowest point that the bottom of the
mass reaches. This is the bottom height, H B . Make
sure that the mass doesn’t hit anything before you
measure the bottom height. Record your measurement
below:
H B = ________________________m
Bottom
Height HB
7. Record the time displayed on the timer, t, below. The
time is in seconds. This is the amount of time that it
takes for the 0.500 kg mass to pass through the timer the
first time it goes through.
t = _____________________________
8. You will also need to know the length of the body of
the 0.500 kg, L, mass to determine the speed in the
middle. Measure L and record your measurement below:
L = _____________________m
- 41 -
Length of
the mass
L
mass
Spring Constant
In order to find the elastic energy, you will first need to find the spring constant of the
spring. The spring constant is the equal to the force exerted by the spring divided by the
amount that it is stretched.
The weight of the 0.500 kg mass is: Weight = mass × grav. acceleration = __________
How big is the force exerted by the spring on the 0.500 kg mass when the mass is
hanging at rest in the middle?
F = _____________________
The amount that the spring is stretched when the mass is hanging in the middle is the
difference of the top and middle heights:
Stretch = H T -H M = ____________________
The spring constant is the force exerted by the spring divided by the stretch:
Spring Const = F/Stretch = _______________________
Energy at the Top
Gravitational: Calculate the gravitational energy, GravE, of the 0.500 kg mass at the top.
GravE = Mass × gravitational acceleration × H T = __________________
Kinetic: What is the speed of the 0.500 kg mass at the moment you release it?
What then is the kinetic energy, KinE, of the mass at the top when you release it?
KinE = ____________________
Elastic: How much is the spring stretched when you release the 0.500 kg mass?
What then is the elastic energy, ElastE, of the spring at the top?
ElastE = ____________________
Total: What is the total gravitational+kinetic+elastic energy at the top?
Top Total = GravE+KinE+ElastE=_____________________
- 42 -
Energy in the Middle
Gravitational: Calculate the gravitational energy, GravE, of the 0.500 kg mass in the
middle.
GravE = Mass × gravitational acceleration × H M = __________________
Kinetic: As the 0.500 kg mass goes through the timer it moves a length, L, in a time, t.
Use the measured values to find the speed (avg. speed) in the middle:
speed = L/t=________________________________
Use the speed and the mass to find the kinetic energy in the middle?
KinE = (1/2) × mass × (speed)2 = ____________________
Elastic Energy: From the stretch and the spring constant you can find the elastic energy:
Stretch = H T -H M = ____________________
ElastE = (1/2) ×(Spring Const) × (Stretch)2 = ____________________
Total: What is the total gravitational+kinetic+elastic energy in the middle?
Middle Total = GravE+KinE+ElastE=_____________________
Energy at the Bottom
Gravitational: Calculate the gravitational energy, GravE, of the 0.500 kg mass at the
Bottom
GravE = Mass × gravitational acceleration × H B = __________________
Kinetic: What is the speed of the 0.500 kg mass at the moment it reaches the bottom?
What then is the kinetic energy, KinE, of the mass at the moment it reaches the bottom?
KinE = ____________________
Elastic Energy: The amount that the spring is stretched when the mass is hanging at the
bottom is the difference of the top and middle heights:
Stretch = H T -H B = ____________________
From the stretch and the spring constant you can find the elastic energy
ElastE = (1/2) ×(Spring Const) × (Stretch)2 = ____________________
Total: What is the total gravitational+kinetic+elastic energy at the bottom?
Bottom Total = GravE+KinE+ElastE=_____________________
- 43 -
Comparisons
According to the law of conservation of energy, the total energy should be the same at the
top, the middle and the bottom. Is the total energy the same at all three places?
Find the percent discrepancy between the Top Total and the Middle Total energies. Use
the Top Total as the Theoretical value.
Middle %D = _______________
Find the percent discrepancy between the Top Total and the BottomTotal energies. Use
the Top Total as the Theoretical value.
Bottom %D = _______________
Given the size of the discrepancies, do you think there is a significant difference between
the total energies at the top, the middle and the bottom?
0.200 kg Mass: Repeat the procedure for the 0.200 kg mass
1. Middle Height:
HM = _________________________m
2. Position the photogate so that mass hangs in the middle of the infrared beam both
vertically and horizontally.
4. Timer Setup is the same
3. Top Height:
H T = _________________________m
5. While holding the 0.200 kg mass at the unstretched position. Reset the time to zero by
pushing the red reset button.
6. Drop the 0.200 kg mass. Measure H B = ________________________m
7. Time on timer: t = _____________________________
8. Length of the body of the 0.200 kg mass: L = _____________________m
- 44 -
Energy at the Top
Gravitational: GravE = Mass × gravitational acceleration × HT = __________________
Kinetic: KinE = ____________________
Elastic: ElastE = ____________________
Total: Top Total = GravE+KinE+ElastE=_____________________
Energy in the Middle
Gravitational: GravE = Mass × gravitational acceleration × HM = __________________
Kinetic:
speed =L/t= ________________________________
KinE = (1/2) × mass × (speed)2 = ____________________
Elastic:
Stretch = H T -H M = ____________________
Elastic Energy: Use the same spring constant as in the previous sections
ElastE = (1/2) ×(Spring Const) × (Stretch)2 = ____________________
Total: What is the total gravitational+kinetic+elastic energy in the middle?
Middle Total = GravE+KinE+ElastE=_____________________
Energy at the Bottom
Gravitational: GravE = Mass × gravitational acceleration × HB = __________________
Kinetic:
KinE = ____________________
Elastic Energy: Stretch = H T -H B = ____________________
Elastic Energy: Use the same spring constant as in the previous sections
ElastE = (1/2) ×(Spring Const) × (Stretch)2 = ____________________
Total: What is the total gravitational+kinetic+elastic energy at the bottom?
Bottom Total = GravE+KinE+ElastE=_____________________
- 45 -
Comparisons
According to the law of conservation of energy, the total energy should be the same at the
top, the middle and the bottom. Is the total energy the same at all three places?
Find the percent discrepancy between the Top Total and the Middle Total energies. Use
the Top Total as the Theoretical value.
Middle %D = _______________
Find the percent discrepancy between the Top Total and the Bottom Total energies. Use
the Top Total as the Theoretical value.
Bottom %D = _______________
Given the size of the discrepancies, do you think there is a significant difference between
the total energies at the top, the middle and the bottom?
- 46 -
II. Conclusion
Write a short paragraph explaining what you learned in this experiment.
What were things you did not understand in this experiment?
- 47 -
Experiment 6
Specific Heat
Theory
The First Law of Thermodynamics
“The total energy of a system remains constant.”
The first law of thermodynamics is a statement of conservation of energy. If an object A
comes into contact with an object B, where object A has a higher temperature (and
therefore more thermal energy), the thermal energy (or heat) gained by one object will
equal the thermal energy lost by the other object. Which object loses thermal energy and
which one gains thermal energy is determined by the second law of thermodynamics.
The Second Law of Thermodynamics
“The entropy of an isolated system not in equilibrium will tend to increase over time,
approaching a maximum value at equilibrium.”
The second law of thermodynamics is a statement of order in the system. Over time,
entropy (or disorder) increases. Heat always flows from a hotter object to a cooler object
because the thermal energy becomes more disordered in this process. The thermal energy
will flow until the system reaches a state of thermal equilibrium at which point the two
objects reach the same temperature. The hot body loses precisely the same amount of
heat as the cold body gains, in accordance with the First Law of Thermodynamics.
The Third Law of Thermodynamics
“As a system approaches absolute zero of temperature, all processes cease and the
entropy of the system approaches a minimum value.”
The minimum value of entropy is zero, although no system can actually reach this state.
As a consequence, no heat engine can ever be 100% efficient.
Specific Heat
The specific heat of a material is defined as the amount of heat we need to add in order to
raise the temperature of a given mass of material by one unit of temperature (normally
1°C or 1K). Because each material has its own specific heat some materials change their
temperature more quickly and easily than others.
Water, for example, has a specific heat of 4200 J/kgoC. This means that you would need
4200 joules of energy to heat one kilogram of water a mere one degree. By way of
comparison, to raise the temperature of one kilogram of aluminum by one degree Celsius
you need only 900 joules of energy, because its specific heat is 900 J/kg°C. Other things
being equal, the different specific heats mean that aluminum will change temperature a
lot faster than water.
- 48 -
object A
hot
+
object B
cool
object A
object B
loses heat
gains heat
=
.
The second law of thermodynamics states that heat will flow from the hotter object
(object A) to the colder object (object B). This process will continue until the two objects
have the same temperature, then the flow of heat will stop. The heat lost by object A will
equal the heat gained by object B as long as no heat is transferred from the two objects to
the environment surrounding the objects.
m A = mass of A
c A = specific heat of A
Q A = heat lost by A
T Ai = initial temperature of A
T Af = final temperature of A
m B = mass of B
c B = specific heat of B
Q B = heat lost by B
T Bi = initial temperature of B
T Bf = final temperature of B
The heat lost, mass of the object, specific heat of the object and the initial and final
temperatures are related by the following formula.
Q A = m A c A (T Ai – T Af )
Object A is losing heat so the initial temperature will be greater than the final
temperature.
Q B = m B c B (T Bf – T Bi )
Object B is gaining heat so the final temperature will be greater than the initial
temperature.
By the first law of thermodynamics, the heat lost equals the heat gained so Q A equals Q B .
QA = QB
m A c A (T Ai – T Af ) = m B c B (T Bf – T Bi )
Since heat will stop flowing when object A and object B have reached the same
temperature, the final temperature of object A and object B will be the same.
T Af = T Bf = T f
This gives us:
m A c A (T Ai – T f ) = m B c B (T f – T Bi )
We will use this relationship to determine the specific heats of different materials.
- 49 -
Your Name ______________________
Lab Partner ______________________
Experiment 6
Work Sheet
I. Specific Heat
First, measure the masses of the empty dry calorimeter (Styrofoam cup), and each of the
three metal samples (lead, copper, and aluminum) and enter the values below. Then put
the three masses in the boiling water with the attached threads hanging out (so that they
can be removed from the water safely). Leave the samples in the boiling water for
several minutes; we can assume that they reach thermal equilibrium with the boiling
water. The initial temperature of the samples will be 100°C.
For Aluminum:
Fill the calorimeter with just
enough cold water to cover
the aluminum sample. Do not
put the aluminum sample in
yet. Measure the mass of the
empty, m cal , and filled,
m cal+water , calorimeter.
m cal = ___________g
water heater
Calorimeter
(foam cup)
m cal+water = ________g
Take this mass and subtract the mass of the empty dry calorimeter and you will get the
mass of the water.
m cal+water – m cal = M water = ___________g
Measure the mass of the Aluminum cylinder, M Al and record the result in the table
Record the temperature of the water in the calorimeter. Add the Aluminum cylinder to
the calorimeter. Gently move the Aluminum cylinder up and down to circulate the water
in the calorimeter being careful not to let the cylinder touch the bottom of the calorimeter
or to let the cylinder come above the surface of the water. Also stir the water gently with
the thermometer. The temperature reading on the thermometer will rise as heat flows
from the Aluminum cylinder to the water. Keep your eye on the thermometer and record
the final temperature. It should take a few minutes to reach thermal equilibrium.
- 50 -
M water (g)
M Al (g)
T water (in cal)
o
( C)
T Al (in boilier)
o
( C)
o
T f ( C)
Aluminum (Al)
When you have recorded all your values, calculate the specific heat C Al for Aluminum.
Compare your calculated values with the theoretical values. Remember that the specific
heat for water ( C water ) is 4200 J/kgoC.
M Al C Al (T Al – T f ) = M water C water (T f – T water )
o
C Al exp (J/kg C)
o
C Al theo (J/kg C)
Aluminum (Al)
%D
900
Repeat the procedure for the copper and the aluminum and calculate C Cu for copper, and
C Al for aluminum. Use fresh water for each object.
For Copper:
m cal = ___________g
m cal+water = ___________g
Take this mass and subtract the mass of the empty dry calorimeter and you will get the
mass of the water.
m cal+water – m cal = M water = ___________g
M water (g)
M Cu (g)
T water (in cal)
o
( C)
T Cu (in boilier)
o
( C)
o
T f ( C)
Copper (Cu)
M Cu C Cu (T Cu – T f ) = M water C water (T f – T water )
o
C Cu exp (J/kg C)
o
C Cu theo (J/kg C)
Copper (Cu)
388
- 51 -
%D
For Lead:
m cal = ___________g
m cal+water = ___________g
Take this mass and subtract the mass of the empty dry calorimeter and you will get the
mass of the water.
m cal+water – m cal = M water = ___________g
M water (g)
M Pb (g)
T water (in cal)
o
( C)
T Pb (in boilier)
o
( C)
o
T f ( C)
Lead (Pb)
M Pb C Pb (T Pb – T f ) = M water C water (T f – T water )
o
C Pb exp (J/kg C)
o
C Pb theo (J/kg C)
Lead (Pb)
%D
130
Discussion: Why do we use a Styrofoam cup instead of a plastic or metal cup?
Discussion: Did your experimental value agree exactly with your theoretical value?
Why not?
Discussion: Was the second law of thermodynamics shown to be true?
- 52 -
II. Conclusion
Discussion: Which element caused the water temperature to increase the most?
According to what you have learned about specific heats, does this make sense?
Write a short paragraph explaining what you learned in this experiment.
What were things you did not understand in this experiment?
- 53 -
Experiment 7
Waves
Theory
Waves
Wave phenomena are observed in a variety of everyday experiences. The most obvious
examples of waves are ripples on the surface of water. You’ve all seen the way the
ripples propagate away from the point where a rock is dropped into the water. There are
many other examples of familiar phenomena which are not as obviously wave related.
The most important of these are sound and electromagnetic waves. Electromagnetic
waves include such things as light, radio, microwaves, infrared radiation, ultraviolet, and
X-rays. When we study waves, we are thinking about some sort of disturbance (like the
rock falling into the water) that propagates away from the point of origin. For sound
waves the source might be a loud-speaker or a musical instrument, or a person’s vocal
chords. For electromagnetic waves the source might be a transmitting antenna (radio), or
a light bulb (light). Waves do not travel infinitely fast; rather, they have a finite,
characteristic speed. Electromagnetic waves move at the speed of light (3 x 108 m/s)
which is extremely fast. Sound moves at the speed of sound which is about 340 m/s. If
you hear echoes, you are hearing evidence of the finite speed of sound. Waves on the
surface of water move at different speeds depending on the wavelengths and depth of the
water.
Transverse and Longitudinal Waves
There are two basic types of wave disturbances: transverse and longitudinal. Transverse
waves are waves in which the direction of the disturbance or movement is perpendicular
to the direction of motion of the wave. An example of a transverse wave would be the
side to side motion of a plucked string. In longitudinal waves, the disturbance is along
the direction of motion. Sound is an example of a longitudinal wave. A sound wave’s
disturbance changes the density of the air through which it moves. As the sound wave
passes, the density increases and then decreases slightly. The compression of the air is in
the direction of motion. At first it would seem that water waves are transverse waves, but
they are actually a combination of the two. In a water wave, the water actually moves in
small circles. So in a water wave there is both movement in the direction of motion and
movement perpendicular to the direction of motion.
Wavelength and Frequency
Waves are usually characterized by their wavelengths and/or their frequencies. Often
waves come in “trains” of peaks and troughs that repeat over and over along the length of
the wave.
- 54 -
Wavelength λ(cm)
How long the wave is.
Amplitude A (cm)
How high the wave is.
The distance from one peak to the next peak, or from one trough to the next trough (in
other words, the distance at which the pattern begins to repeat itself) is called the
wavelength. We often use the Greek symbol, lambda λ , to represent the wavelength.
Another characteristic of the wave is its frequency. Imagine that you are floating on the
surface of the water as waves pass by. As you float you move up and down with an
oscillating motion. This oscillation has a characteristic period, which is the amount of
time it takes to go through one oscillation. Another way to quantify the rate of the
oscillation is the frequency of oscillation. The frequency is the number of oscillations
that take place in a given amount of time. We will use f to represent frequency. The
frequency is the reciprocal of the period.
f = 1/Period
In other words, if the period of oscillation is 2.0 seconds (2.0 s) then the frequency is 0.5
s-1 or 0.5 Hz. The unit of frequency is called the Hertz (Hz). 1 Hz = 1 s-1. If you look at
a radio tuner, you will notice that the frequencies of AM radio broadcasts range from
about 540 to 1500 kHz, where a kHz = 1000 Hz. FM radio broadcasts range in frequency
from about 90 MHz to 110 MHz, where one MHz is 1,000,000 Hz. You can see that
radio frequencies are very high. Audible sound waves have much lower frequencies.
People can hear sounds within the frequency range of about 20 Hz to 20,000 Hz. As you
get older your ability to hear the higher frequencies diminishes; most adults cannot hear a
tone of 20,000 Hz. The pitch of a musical note corresponds to the frequency of the sound
wave. For instance, middle C on the piano corresponds to a frequency of about 260 Hz.
The first A above middle C corresponds to a frequency of 440 Hz.
The wavelength and the frequency are related to the wave speed. This can be shown
easily in the following illustration of a moving water wave and a stationary buoy.
- 55 -
X
v
X
Time: 0
v
X
Time: 0.5T
v
Time: T
λ
As you can see, the wave moves one wavelength, λ , during one oscillation period, T.
Consequently, the wavelength must be equal to the product of the wave speed, v and the
period.
λ = vT
Since the frequency, f, is the reciprocal of the period, T, the wave speed must then be
equal to the product of the wavelength and the frequency
λf = v
Waves of many different frequencies can be present in the same place at the same time.
For example, often in music there is more than one note being played at a time. The
radio waves from each of the different available radio stations are passing through the air
at the same time. Your tuner just picks out the one that you choose and filters out all the
others.
When waves of different frequencies or waves traveling in different directions meet, their
amplitudes simply add together. This property is called superposition. For instance a
- 56 -
high frequency wave and a lower frequency wave traveling together might look like the
illustration below.
Two waves of the same wavelength traveling in opposite directions also add together this
way. Often waves become trapped in a region because they reflect off the boundaries of
the region. You have all seen how water waves reflect off the sides of a swimming pool,
and you have all heard an echo. These are both examples of reflection. Reflection
doesn’t occur only where there is a hard boundary inhibiting the passage of the waves. It
can also occur where there is an abrupt change in the medium that is transmitting the
wave. An example of this is the reflection you see in a transparent piece of glass. This
reflection occurs because the speed of light is slower in the glass than in the air.
Standing Waves
Incident wave
Reflected wave
Another type of wave that can be made is the deflection on a stretched string. You may
have noticed that if you stretch a rope tight and then move one of the ends quickly, it
sends out a ripple much like the ripple on the water. If you stretch the string tightly
between two fixed points, waves traveling along the string will be reflected back along
the string in the other direction.
Constructive interference
Destructive interference
These reflected waves will then be added to the oncoming waves, and they will continue
on to be reflected back at each end. This addition of waves is called interference.
There are certain wavelengths at which the waves traveling in the two directions will add
together coherently. When this happens, there are some points on the string, called
nodes, where they will always cancel each other out exactly. At a node, the string is
- 57 -
stationary. There are other places where the waves will add together to produce an
oscillation twice as large as the individual waves would produce. We call these points
anti-nodes. When such waves are made on the string it is called the condition of
resonance. Remember that this condition depends on the wavelength; it is equivalent to
say that it depends on the frequency. Such a wave is called a standing wave, and the
frequency used to make it happen is called a resonant frequency. The string stretched
tight between two fixed points constitutes a resonant cavity. Other examples of resonant
cavities are organ pipes and wind instruments in general. These are examples of acoustic
(sound wave) resonant cavities. A guitar string is another example of a resonant cavity.
The inside of a laser tube is an optical resonant cavity.
For the stretched string, the condition for resonance is that the length of the string, L, be
some multiple of twice the wavelength, λ .
`
λ = 2L/n
(n = 1, 2, 3, 4 …)
Some of the standing waves on a stretched string would look something like this:
λ= 2L
n=1
L= resonant cavity length (cm)
λ= L
n=2
node
λ= 2L/3
n=3
antinode
λ= L/2
n=4
As shown in the illustrations above, standing waves do not move back and forth. Instead,
the standing waves oscillate back and forth in place, but the amplitude is different at
different points on the string. Remember that what we mean by amplitude is the size of
the disturbance.
- 58 -
Your Name__________________
Lab Partner _________________
Experiment 7
Work Sheet
In this experiment we will generate some standing waves on a stretched string, we will
measure some resonant frequencies of the stretched string, and we will see how changing
the tension and the length of the string change each of the resonant frequencies.
I. The Apparatus
Attach the 200 g mass to the end of the string. Pass the string over the pulley and attach
the other end to a fixed point on the lab stand.
Resonant cavity length L
wave driver
tension on string (T)
**
frequency
signal generator
m
1000.0
frequency
knob
gravitational force ( Fg=mg)
amplitude
knob
Fg=T
Connect the signal generator to the wave driver with the two cables provided. Attach the
drive rod of the wave driver to the string. Measure the length of the stretched string, L 1 ,
from the drive rod to the pulley.
L 1 = ___________
II. Standing Waves
Turn the signal generator on. Decrease the frequency until you find a resonant mode.
Record the frequency displayed on the signal generator and the corresponding resonant
mode number, n, in the following table. Repeat this for the other modes of the string
shown above.
- 59 -
Trial A
Tension = .200 kg × 9.8 m/s2 = 1.96 N
Length = L 1
λ (cm)
n
f (Hz)
v (cm/s)
Repeat the experiment with a different tension on the string by replacing the 200 g mass
with the 500 g mass. This changes the tension on the string from 1.96 N to 4.9 N.
Trial B
Tension = .500 kg × 9.8 m/s2 = 4.9 N
Length = L 1
λ (cm)
n
f (Hz)
v (cm/s)
Repeat the experiment with a different resonant cavity by shortening the string to half the
original length. Replace the 500 g mass with the 200 g mass to bring the tension back to
the original 1.96 N.
L 2 = ___________
Trial C
n
Tension = .200 kg × 9.8 m/s2 = 1.96 N
Length = L 2
λ (cm)
f (Hz)
v (cm/s)
Discussion: How does the tension affect the velocity of the wave? How does the length
affect the velocity of the wave?
- 60 -
III. Conclusion
Discussion: When you tighten (increase the tension) a guitar string, its frequency of
vibration (resonant frequency) increases. Is this consistent with the results of this
experiment (Trial A vs. Trial B)? Explain.
Discussion: Musical instruments with shorter lengths usually make higher pitches. Is
this consistent with the results of your experiment (Trial A vs. Trial C)? Explain.
Write a short paragraph explaining what you learned from this experiment.
Was there anything that you did not understand about this experiment?
- 61 -
Experiment 8
Electricity
Theory
Electricity
Electric power is such an integral part of our lives that most of us could not imagine life
without it. So it is important that we understand some of the basic principles of
electricity. Electricity involves the flow of charged sub-atomic particles called electrons
through wires or other conductors. Electrons are too small to be seen, and in general, the
flow of electrons is invisible, but electric power provides the energy that runs electric
motors, and all other electric appliances. In addition to energy to move things around
electricity provides the energy that is converted into light in light bulbs. Electricity also
allows us to transmit and receive electromagnetic signals (radio and TV waves), and
convert them back to sound and pictures. Inside a television or computer monitor, other
electrical devices use more electrons to “paint” pictures on the tube. The extremely small
and fast arrays of switches (chips) that are the heart of modern computers, cellular
phones, calculators, engine controls in automobiles, etc. are also examples of electric
circuits.
Electromagnetic Fields
Electrons have a special property. We say that they have electric charge. Charged
particles interact with each other (exert forces on each other) through electromagnetic
fields. Electromagnetic fields cause forces that act at a distance, much like the
gravitational force of the earth pulls on the moon. There are two kinds of
electromagnetic forces. The first is the electrostatic force. When two electrons are
brought close together, they repel each other. The electric charge of a particle can be
either negative or positive. Electrons actually are negatively charged. Other particles
(for instance, protons) have positive charges. Two particles with opposite charges attract
each other, and two particles with like charges repel each other. Because of this
attractive force, protons and electrons tend to group together in atoms in equal numbers,
which means that atoms have no net charge. This generally cancels out the effect of the
individual charges on more distant charged particles. However, you experience the
electric repulsion of the electrons in a solid object whenever you touch it and your hand
does not pass through it.
The second kind of electromagnetic force is the magnetic force. Whenever charged
particles are in motion, they generate a magnet field. The action of the magnetic force is
more complex than that of the electric field, but it bears some similarities to the electric
field. If you have ever played with a bar magnet, you may have noticed that it has two
poles, a north pole and a south pole. If you bring together two bar magnets, the opposite
poles attract and the like poles repel much like the charges in the electrostatic case.
Electromagnetic forces are used extensively in our everyday lives. They are used in the
electric motors in many electric appliances. They are used to drive loud-speakers.
- 62 -
Electromagnetic forces are used to convert the mechanical motion produced by
hydroelectric, fossil fuel burning, and nuclear power plants into electricity.
Electric Circuits
The operation of any electrical device requires that there be a flow of electrons, which we
call an electric current. In order for electric current to continue, it must flow in a loop;
otherwise, electrons will build up at some point, where they will resist the continued flow
of electrons. A very simple electric circuit is pictured below.
In this circuit, the battery provides
the power that keeps the electrons
frequency
flowing around the circuit. The
lines represent wires, which carry
the electric current. The electric
power is converted into light in the
light bulb. The electrons in the
light bulb filament collide with the
atoms of the filament and lose
current (I)
some of their energy by heating up
the filament. The battery converts
chemical energy into electric
power and that power is then
converted into light energy in the
light bulb. In order for the circuit to continue to work, the flow of electric current around
the circuit must be maintained. If you break the loop, the current will stop flowing and
the light bulb will stop glowing.
Volts and Amperes
Even though the circuit described above is very simple, all electric circuits work on the
same principles. There must be a flow of electrons around a loop, and there must be a
source of electric power, like the battery. Most electric circuits use a battery or the power
provided by the electric power company through the electric outlets in the house or
building. In an electric circuit there are two basic variables: current and voltage. The
current is the rate of flow of electric charge, measured in units of Amperes or Amps (A).
The voltage is a measure of the potential of the power supply for providing electric
power. The voltage is measured in units of Volts (V). There are instruments for
measuring electric current and voltage just like there are instruments (like meter sticks
and scales) for measuring length and mass. Voltages are measured with voltmeters, and
currents are measured with ammeters.
Alternating and Direct Current
- 63 -
Batteries are designed to provide a constant voltage. For instance, a standard flashlight D
cell produces a voltage of about 1.5 Volts. However, the voltage provided by a wall
outlet is not constant. In fact, voltage provided by the wall outlet is oscillating at a
frequency of about 60 Hz. It oscillates between a maximum of about 120 V to a
minimum of about -120 V. The power company provides power in this form because less
of it is lost along the power lines this way. When the voltage and current oscillate this
way it is called Alternating Current (AC). When electric power comes in the form of a
constant voltage and current it is called Direct Current (DC).
Ohm’s Law
Electric circuit elements (like our light bulbs) all resist the flow of electric current. Some
offer more resistance than others. The more resistance the element has, the less current
will flow (at a certain voltage). Electric circuit elements can be characterized by their
resistance, R, which is the ratio of the voltage applied, V, to the current through the
element, I:
R = V/I
Some types of circuit elements have resistance which is independent of the particular
value of the voltage or the current. These circuit elements are called resistors. Resistors
follow Ohm’s law, which states that the current through the resistor is directly
proportional to the voltage. That is, a resistor’s resistance is constant. Some other
resistive elements do not follow Ohm’s law. In other words, their resistance, (the ratio
V/I), is not constant. One example of this behavior is a light bulb. As the filament in the
light bulb gets hotter, it’s resistance to the electric current increases.
Electric Power
Electric circuits convert electric power into other forms of energy including mechanical
energy (motion), thermal energy, and light. The total amount of energy is conserved
(never changes) so this energy must come from somewhere. This energy comes from the
source of electric power, the battery or electric generating plant. The electric circuit is
said to “consume power.” In fact, the circuit simply converts energy from one form to
another. The amount of power “consumed” by a circuit element, P, is equal to the
product of voltage applied to it, V, and the current through it, I.
P = VI
We measure power in units of watts. A watt is a unit of energy converted per unit time.
The metric unit of energy is called a joule (J). The joule is related to the units of mass,
length and time.
1 J = 1 kg m2/s2
- 64 -
A watt is a Joule/sec. For example, a 100 watt light bulb consumes (converts to light
energy) 100 joules of energy per second. If the bulb has an average of 100 V applied to
it, then it has about one amp of current flowing through it on average.
For a given voltage, V, the more current that flows, the more power will be consumed.
Consequently, elements that have greater resistance consume less power. In fact, the
power consumed by a resistor is inversely proportional to the resistance, R.
P = V2/R
Parallel and Series Circuits
Most circuits consist of more than one element. A simple example of an electric circuit
with more than one element is a set of Christmas tree lights. There are two basic ways
that several elements can be connected. The elements can be wired in series one after
another as shown below.
Series
Circuit
There is a problem involved with wiring lights this way. If one of the lights burns out,
the circuit is broken. If the circuit is broken, it no longer functions. When circuit
elements are wired this way, each element add a little resistance to the total. So all the
lights wired together are more resistant to current than one alone. In fact, the resistant of
the whole string of lights is equal to the sum of the resistances of the individual lights.
𝑅𝑆𝑒𝑟𝑖𝑒𝑠𝑇𝑜𝑡𝑎𝑙 = 𝑅1 + 𝑅2 +∙∙∙ +𝑅𝑛
- 65 -
The other way that you can combine more than one element in a circuit is in parallel.
When several elements are connected in parallel, both of the leads from each element is
connected by wire directly to the power supply as shown below.
Parallel
Circuit
One advantage of connecting lights in parallel is that one lamp burning out will not
interrupt the rest of the circuit. If one lamp burns out, it only breaks the current in that
branch of the circuit. Current can still flow through the rest of the circuit. When several
elements are connected in parallel, the amount of current being drawn from the power
supply is more than would be drawn by any one of the individual elements alone.
Consequently, connecting several elements together in parallel reduces the effective
resistance of the entire circuit to less than any of the individual elements alone.
1
𝑅𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑇𝑜𝑡𝑎𝑙
=
1
1
1
+
+∙∙∙ +
𝑅1 𝑅2
𝑅𝑛
- 66 -
Your Name _________________
Lab Partner _________________
Experiment 8
Work Sheet
I. Current, Voltage and Resistance
You will determine the resistance of a resistor. Use the batteries as the power supply
(DC). Connect the battery holder, the resistor, the voltmeter, and the ammeter as shown.
A voltmeter measure the voltage across a circuit element and an ammeter measures the
current flowing through the circuit element.
ammeter
-
0
multimeter
-
+
+
+
4 2
resistor
-
3
+
1
Beginning with one battery: read the current and voltage across the resistor from your
meters. Record the data in the table. Repeat with 2, 3, and 4 batteries. The current is
measured in milliamps so make sure you convert to amperes (divide by 1000). Calculate
the resistance of the resistor using R=V/I.
# batteries
Voltage (V)
Current (A)
Resistance (Ohms)
1
2
3
4
Discussion: The resistor is ohmic, which states that the resistance is constant. Is the
resistance in your table roughly constant?
- 67 -
II. Electric Generators
We will explore the phenomenon of electric power using the concept of induction in the
hand held mini-generator. The mini-generator works much the same way as a
conventional electric generator. When you turn the hand crank it rotates a coil of wire in
the magnetic field of a permanent magnet inside the generator. If there is an unbroken
circuit the current will then flow out through the wire leads coming out of the generator.
mini-generator
wire leads
light bulb
This creates a complete circuit with the generator as the power supply. Current will flow
from the generator into the light bulb where it will heat up the filament until it glows.
Connect the light bulb to the generator as shown. Turn the crank of the generator until
the light glows. If it doesn’t glow at first, turn the crank faster until it does. Notice that
the fast you turn the crank the brighter the lamp glows.
Discussion: Why do think this happens?
Now disconnect one of the wire leads and begin turning the crank as before
Discussion: What happens?
- 68 -
Now connect two of the mini-generators as shown below.
Turn the crank on one of the generators.
Discussion: What happens to the other generator when you turn the crank on the first?
How could you explain what is happening?
III. Series and Parallel Circuits
Look at the pictures below. Label below each picture which is a series circuit and which
is a parallel circuit?
___________________
_________________
Use the multimeter; set the multimeter to the 200 Ohms scale to measure the series
resistors and the parallel resistors. Knowing that each resistor has a value of 50Ω. Find
the theoretical total effective resistances and calculate the percent discrepancies from
your experimental values.
- 69 -
Rexperimental(Ω)
Rtheoretical(Ω)
% Disc.
Series
Parallel
Discussion: What happens to the total effective resistance as you increase the number of
resistors in series?
Discussion: What happens to the total effective resistance as you increase the number of
resistors in parallel?
- 70 -
VII. Conclusion
Write a short paragraph describing what you learned from this experiment.
Was there anything about this experiment that you didn’t understand?
- 71 -
Experiment 9
Electromagnetism
Theory
The Electromagnet
When a current passes through a wire it produces a magnetic field similar to the field
produced by a permanent magnet. This magnetic field will exert a force on other
magnets and on objects containing iron or other ferromagnetic materials. You can
demonstrate this by bringing the wire close to a compass and noting that the presence of
the wire with current running through it can change the direction that the compass points.
Normally, the compass points along the direction of the magnetic field of the Earth (the
Earth itself is a giant magnet), but when another magnetic field of equal or greater
strength is introduced it can change the direction along which the compass points. The
compass needle is a bar magnet suspended at its center of gravity and allowed to turn
freely. A magnet which is allowed to turn freely will tend to align itself with the
magnetic field lines of the magnetic field present (if there is any magnetic field present).
The magnetic field lines do not represent the direction of the magnetic force, but they do
represent the direction that other magnets (like the compass needle) will tend to point
under the influence of a magnetic field.
bar magnet
current
wire
As shown to the left, a current carrying wire
produces a magnetic field much like a permanent
magnet. We can increase the strength of the field
produced by wrapping the wire into a coil as
shown below. When the wire is wrapped in a coil
the magnetic fields from each of the turns of the
coil add together in the same direction, thus
increasing the strength of the field.
magnetic
field lines
current carrying coil
bar magnet
The magnetic field lines from the coil of
wire resemble the shape of the field lines
produced by the permanent magnet.
current
Another way to strengthen the magnetic
field of the coil of wire is to introduce an
iron (or other ferromagnetic) core to the
coil. The atoms in the iron core have
spinning and orbiting electrons which
make each atom a tiny magnet.
N
S
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When a magnetic field is introduced from outside
the iron, as it would be from the coil of wire, the
atoms begin to line up with the field lines just like
the compass lines up with Earth’s field lines.
Before the field was introduced, the atoms were all
lined up in random directions so there was no net
current
magnetic field present, but as they line up in
response to the applied magnetic field of the coil,
the magnetic field of each of the atoms adds to the
applied magnet field of the coils. Even though the
contribution of each atom is very small the net
result of all of them lining up is quite large. The
end effect is that the coil with the iron core is a
c
much stronger magnet than the coil alone. The
coil of wire used as a magnet is called an
electromagnet and most electromagnets make use
of the iron core to increase the strength of the
magnetic field.
iron core
If you turn off the current in the electromagnet
with an iron core, the magnetic field will decrease but not all the way to zero. Many of
the atoms in the iron core will remain aligned in the same direction. Consequently, the
core of the electromagnet is now a permanent magnet. This is how permanent magnets
are manufactured.
N
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
Electromagnets are very useful tools. Virtually all electric motors use electromagnets to
turn electrical energy into motion. Loud-speakers use electromagnets to turn electrical
signals into sound, and there are many other applications of electromagnetism.
Induction
Induced current
In the electromagnet, the magnetic
field produced by the current in a wire
can be used to cause motion through
the magnetic field produced. The
reverse is also true. The motion or
change of a magnetic field can be
used to drive an electric current. If
you change the magnetic field through
a loop of wire by moving a bar
magnet near to the wire for instance,
it will produce a current in the wire.
S
N
S
N
NS
The change of the magnetic field lines through the loop of wire exert a force on the
electrons in the wire causing them to move. You can also produce a current by moving
the coil near the magnet. Any change in the flux of magnetic field lines through the loop
of wire will induce a current in the wires. This phenomenon is called induction. An
- 73 -
electric generator uses induction to produce electric power by moving a coil of wire in an
magnetic field. In fact, almost all electric power is produced by electric generators
through induction. The mechanical energy might be produced by water falling in a
hydroelectric plant or by steam turbines run from a fossil fuel burning of nuclear power
plant, but all of them use electric generators to produce electricity.
The Transformer
We use induction to convert mechanical energy into electrical energy. Another use of
induction is the transformer. The transformer allows us to transfer electric power from
one electric circuit loop to another through a magnetic field. The transformer consists of
two separate loops of wire wound around the same iron core as shown below. It is
important to not that the two loops of wire are not in contact with each other.
For reasons similar to the ones
discussed for the electromagnet,
top beam of iron core
almost all the magnetic field lines
are confined to the iron core. Thus
the iron core serves as a conduit
making sure that almost all the
V1
V2
magnetic field lines produced by
current in one coil go through the
second. In the transformer, a
ind
changing electric current is driven
through one of the coils. This
N2=3
N1=4
number
of
changing current produces a
number of
coils
changing magnetic field in the iron
coils
core, thus changing the flux of
magnetic field lines through the second coil of wire. This change in magnetic flux
induces a current in the second coil much like the motion of the coil in a magnetic field
produces the electric current in the generator. However, if the current in the first coil is
constant (DC), there will be no current induced in the second coil. Transformers only
work with alternating current (AC).
Iron core
Transformers are commonly used to “step down” the voltage in an electric circuit. For
instance, many commonly used electric products require power supplies which deliver
voltages in the range of 9 to 12 Volts, but the electric power supplied through the wall
outlets has a peak amplitude of about 120 Volts. In a transformer, the peak amplitude of
the AC voltage produced in the second coil, V2, is related to the peak amplitude of the
AC voltage applied in the first coil, V1, in that the ratio of the peak amplitudes is equal to
the ratio of number of turns of wire in the coils, N1 and N2 respectively.
V2/ V1 = N2/ N1
In most electronic appliances, the AC voltage is converted into DC voltage after being
“stepped down” by the transformer. This accomplished using a rectifier circuit.
- 74 -
Your Name__________________
Lab Partner _________________
Experiment 9
Work Sheet
I. Magnets
Discussion: Take one of the bar magnets and bring it close to the compass. First move
the north pole of the magnet (the end with the small groove cut near it) near the compass.
What happens?
Discussion: Now move the other end (the south pole) of the bar magnet near the
compass. What happens?
Discussion: Move the compass around the perimeter of the bar magnet keeping the
compass horizontal. What happens?
The Compass and the Earth’s magnetic field
The compass needle is also a bar magnet. It is balanced at its center of gravity so it is
free to turn in any direction. The compass needle will turn until it points along the
direction of the magnetic field lines in its vicinity. In the absence of a strong magnet of
some kind, it will point along the magnetic field lines of the Earth. The Earth acts like a
giant bar magnet with one pole near the north geographic pole and the other near the
south geographic pole. The Earth’s north magnetic pole is actually somewhere in
northern Canada. The north pole of the compass needle is deflected by the Earth’s
magnetic field so that it points toward the north magnetic pole of the Earth. Since the
north magnetic pole is not exactly on the north geographic pole, the compass does not
always point toward true north. It points toward magnetic north which is usually a few
degrees away from true north. Since the north pole of a magnet is defined as the pole that
- 75 -
is attracted toward the north, if the Earth were labeled as a bar magnet, the Earth’s north
magnetic pole (the one near the north geographic pole) would actually be labeled south
and vice versa. This fact can make things somewhat confusing.
Discussion: Take the two bar magnets and bring the two north poles near each other.
What happens? Repeat this with the two south poles. What happens?
Discussion: Bring the south pole of one magnet near the north pole of the other. What
happens?
Bring the bar magnet near the iron transformer core. Notice that the iron core is attracted
to the magnet, but the brass and aluminum bolt that holds it together is not attracted to the
magnet.
Discussion: Are all metals attracted to a magnet?
- 76 -
II. The Electromagnet
Use four batteries in the battery holder and connect them with the 800 turn coil to form a
circuit as shown below.
4
2
3
1
bar magnet
coil
compass
Discussion: Move the compass around the coil from one end to the other. What happens
to the compass needle?
Discussion: Move the compass around the perimeter of the bar magnet keeping the
compass horizontal. How does the compass’ behavior near a bar magnet compare to its
behavior near the coil with a current flowing through it?
Discussion: Take the bar magnet and insert the north pole of the magnet (the side with
the notch on it) into one end of the coil. Describe what happens as you push the bar
magnet through the coil.
- 77 -
Discussion: Turn the bar magnet around and push the south end into the same end of the
coil as before. Describe what happens.
Remove the top beam of the iron transformer core. Notice that it is not attracted to the
rest of the transformer core. Insert the top beam of the transformer core into the coil.
Leaving the top beam in the coil, bring the rest of the transformer core near the top beam.
Discussion: What happens?
Discussion: Remove the top beam from the coil. Now touch the top beam to the rest of
the transformer core. Is the magnetic attraction as strong as it was with the top beam
inside the coil?
III. Induction
-
0
Connect the 800 turn coil to the galvanometer as
shown.
+
galvanometer
-
1
2
-
coil
3
+
The galvanometer is a very sensitive ammeter.
It is capable of measuring extremely small
currents flowing in either direction through
itself. In order for the galvanometer to work you
must press one of the buttons on its front face.
In this experiment, press the button labeled 1 to
measure the current.
Press the button on the galvanometer and notice
that normally there is no current moving through
the coil.
- 78 -
Discussion: Keeping the button pressed, insert the bar magnet into the coil and move it
in and out. What happens to the current in the coil?
Discussion: Remove the magnet from the coil and shake it vigorously outside but near
the one end of the coil while still pressing button one. What happens to the current in the
coil?
Discussion: Hold the magnet steady and shake the coil vigorously near one end of the
magnet (still pressing the button). What happens to the current in the coil?
IV. The Transformer
Slip two of the coils around the transformer core as shown. Connect one of the coils to
the signal generator with the wires provided. Connect the other coil to a light bulb.
signal generator
2.0000
Turn the amplitude knob on the signal generator all the way down before you turn it on.
Turn the signal generator on and turn down the frequency to about 2 Hz.
- 79 -
Discussion: Turn up the amplitude until you can see the lamp glow. Describe what you
see. (Note that there is no conductive contact between one side of the transformer and
the other.)
Discussion: Disconnect the transformer coil from the signal generator and then connect
it up to the batteries. Does the lamp glow? Why not? (You might convince yourself that
the batteries are capable of lighting the lamp by connecting them directly.)
Turn the signal generator amplitude all the way down. Reconnect the transformer to the
signal generator and the light bulb as before but reverse the roles of the two coils.
Discussion: Turn up the amplitude of the signal generator until the light burns with the
same brightness as before. Is the amplitude of the signal generator bigger or smaller than
before? Can you explain this in terms of the “stepping down” of the voltage described in
the theory section?
- 80 -
V. Conclusion
Discussion: As mentioned earlier the Earth behaves like a giant bar magnet. Can you
think of any way the Earth might have become such a magnet?
Write a short paragraph explaining what you learned from this experiment.
Was there anything that you did not understand about this experiment?
- 81 -
Experiment 10
Light – Reflection and Refraction
Theory
Light
Light is one of the most important phenomena we experience. The sense of sight is the
ability to perceive the light coming from our environment. Anything that we perceive
through sight is the result of light emitted by or reflected from that thing. For most of us,
our eyes are our primary tool for understanding the world around us. The scientific
understanding of the nature of light has developed quite a bit over the years. Isaac
Newton believed that light consisted of tiny particles, which he called corpuscles. Later,
in the nineteenth century, experiments showed that light behaves like a wave. In the
twentieth century, it has been demonstrated that there are other circumstances when light
does behave as if it were made of particles. Physicists call this double nature of light the
wave/particle duality. It has also been shown that matter exhibits a similar behavior.
These peculiar behaviors of light and matter (especially at small scales) gave rise to a
new theory of the nature and behavior of matter and energy called Quantum Mechanics.
However, our experiment will not involve trying to understand light at this level.
light rays
source
In this experiment with light we will use the fact
that in many practical situations light can be
thought to travel along rays. Light rays originate
at a light source such as the sun or a light bulb.
The light rays travel away from the light source
at the speed of light (3×108 m/s), which is very,
very fast. In fact, it is so fast that you cannot
perceive that it takes any time for light to travel.
In general, the light rays will travel in straight
lines away from the source through space or
through the air (radial outward).
However, whenever a light hits a solid or liquid matter, something else will happen. The
light will be absorbed, reflected or transmitted, or some combination of the three. An
object that absorbs practically all the light incident on it is called black. Some things
absorb some colors of light but reflect others. Green objects reflect more green light than
other colors. Red objects reflect more red, etc… White objects reflect all colors equally.
The different colors correspond to the different wavelengths of light. Some objects, like
glass, transmit almost all the light that hits them. Such an object is said to be transparent.
Absorbed
Reflected
- 82 -
Transmitted
Reflection
Most reflective surfaces, like white paper, are very rough at small scales. Consequently,
light rays that hit the surface are reflected in all directions. Such surfaces are called
diffuse reflectors, but a polished metal surface, such as those used in mirrors, is not so
rough. Such a surface will reflect all the light rays incident from one direction the same
way. As a consequence, the reflected light rays from the mirror form an image which
appears to be behind the mirror.
The image behind the mirror is formed as a
consequence of the law of reflection. The law
of reflection states that the angle that they
reflected ray makes with the normal to the
surface of the mirror is equal to the angle that
the incident ray makes with the normal to the
surface, and that all three (the incident ray, the
reflected ray, and the normal) are in the same
plane. The normal to the mirror surface is an
imaginary line that is perpendicular to the
mirror surface at the point where the incident
ray strikes.
Mirrored
surface
The Law of Reflection
object
image
As is shown in the drawing, the law of reflection causes the incident rays from the light
bulb to be reflected in such a way as to appear to be coming from an image behind the
mirror.
Refraction
normal
When a light ray is incident on a transparent surface, it passes through the surface, but it
doesn’t go straight through the surface. The light ray is bent as it passes through. You
may have noticed that when you partially submerge a stick in clear water it appears to be
bent at the point where it enters the water even if it is really perfectly straight. This
apparent bending of the stick is the result of refraction of the light rays that are being
reflected off the stick.
The bending of the light
rays as they pass from the
Incident ray
reflected ray
water into the air causes
the stick to appear to be
slanted at an angle
Өi
Өr
different from the one
with which it enters the
water
mirrored surface
- 83 -
refracted ray
straw
incident ray
air n=1
water n=1.33
end of straw
appears at this
position
real end of straw
Index of Refraction
The refraction of the light rays as they pass through a transparent surface is due to the
difference of the speed of light in the two regions on either side of the surface. Light is
slowed down when it passes through a material. In general, the more dense the material,
the more the light is slowed. The speed of light in air is practically the same as it is in
empty space. However, the speed of light is water is significantly less than the speed of
light in empty space. The ratio of the speed of light in empty space to the speed of light
in a transparent material is called its index of refraction. The index of refraction of water
is about 1.33. The higher the index of refraction of the material, the slower the speed of
light will be within the material. We usually use the letter, n, to represent index of
refraction in equation. Other transparent materials like glass can have much higher
indices of refraction. Different kinds of glass have different indices of refraction.
Water
Air
Typical Glass
Speed of light
=3 x 108 m/s
n=1
Speed of light
=2.3 x 108 m/s
n=1.33
Speed of light
=1.5 x 108 m/s
n=2
Snell’s Law
The amount by which a light ray is bent when it passes through the surface of a
transparent material follows a simple law called Snell’s law, which relates the angle that
the incident ray makes with the normal to the surface (angle of incidence θi), the angle
that the refracted ray makes with the normal to the surface (angle of refraction θr), the
index of refraction of the material on the side of the incident ray, ni, and the index of
refraction of the material on the side of the refracted ray, nr.
Snell’s law states:
ni sinθi = nr sinθr
- 84 -
surface
Sinθ is the trigonometric function that
relates an angle in a right triangle to the
length of the opposite side and the length
of the hypotenuse.
ni
nr
Өr
normal
Өi
hypotenuse
opposite
Ө
sin Ө=opposite/hypotenuse
Lenses
We make use of the refractive properties of glass and other transparent materials in
lenses. A lens is a piece of glass or other transparent material with a curved surface.
Because of the curvature of the lens surface, light is refracted by different amounts at
different parts of the lens. As a consequence of this differential bending, the lends can
focus rays of light that wouldn’t otherwise come together.
Converging and Diverging Lenses
There are two basic types of lenses,
converging and diverging lenses.
Converging lenses bend light rays
inward. Diverging lenses on the
other had bend the light rays outward
away from the center. Each of the
pictures represents a cross section of
a usually circular lens.
Converging/convex lens
You can tell a converging lens from a
diverging lens in that the converging lens
is thick in the middle and thin at the edges,
whereas the diverging lens is thin in the
middle and thick at the edge.
Diverging/concave lens
Images
Lenses are commonly used to form
images. For instance, an overhead
projector uses a converging lens to form
an image on the screen of the transparency on the projector. Movie projectors and slide
projectors project images onto a screen in the same way.
- 85 -
screen
An image that can be projected
onto a screen is called a real
image.
object
A projector is used to project a
large image of a small object
onto a screen. In a camera, the
real
reverse is true. The camera lens
image
makes a small image of a larger
object on the film of the camera.
The chemicals in the film are
sensitive to light and change their transparency to different colors of light depending on
how much light is incident on them. In a video camera, a charged coupled device (CCD)
is used instead of film. A CCD converts the light into electrical signals which are stored
on the video tape.
virtual
image
A converging lens can be used to
make another type of image. If
you hold and object close to a
converging lens and look through
the lens at the object, the object
will appear larger than it really is.
When used this way the lens is
called a magnifying glass.
object
When used this way, the lens
forms a virtual image. The virtual cannot be projected onto a screen. You must look
through the lens itself. When you look through the lens, what you see is the virtual image
and not the object itself. In the magnifying glass, the virtual image is larger than the
object thus making it easier to see small details in the object.
Diverging lenses can form virtual images as well, but the virtual images formed by
diverging lenses are smaller than the object. Diverging lenses cannot, by themselves,
form real images.
The Eye
Your eye makes use of a converging lens to focus light onto the retina which covers the
inside surface of the eye. The focused light forms an image of what is in front of your
eye in much the same way that an image is formed on the film in a camera. The retina
then takes the focused light and converts into neural signals which are transmitted to the
brain through the optic nerve.
- 86 -
lens
retina
image
focus
optical
nerve
Many people have eyes that cannot form a clear image without corrective lenses. This
results from the fact that the image formed by the lens does not come into focus on the
retina. In near-sighted people, the image of distant objects comes into focus in from of
the retina rather than on the retina as in a person with good vision, and in far-sighted
people the image of near by objects comes into focus behind the retina. Consequently,
near-sighted people cannot see distant objects clearly and far-sighted people cannot see
near by objects clearly.
Farsightedness
Nearsightedness
Corrective eyeglasses or contact lenses are used to correct these deficiencies in the ability
of the eye to focus. For near-sighted people, the corrective lens is a diverging lens. For
far-sighted people, the corrective lens is a converging lens.
corrected
farsightedness
corrected
nearsightedness
- 87 -
Optical Instruments
In addition to projectors and cameras, there are many other uses for lenses. The most
common optical instruments are telescopes and microscopes.
Telescopes are used to make distant objects appear nearer. A simple telescope consists of
two lenses, an objective lens and an eyepiece. The objective lens forms a real image of a
distant object. The eyepiece converts that real image into a virtual image that appears
much closer than the object really is, in a simple telescope the image appears upside
down.
bottom ray exits as top ray
Astronomical telescopes are built with large objectives so that they can gather as much
light as possible in order to see very faint stars and galaxies. Usually, the objective of an
astronomical telescope is a mirror, not a lens. Mirrors can be used to focus light in much
the same way as lenses are used to focus light. The objective mirror of the Keck
telescope on Mauna Kea in Hawaii is ten meters in diameter.
eyepiece
secondary
mirror
primary
mirror
The microscope is an optical instrument that is used to make small things appear larger.
The simple microscope consists of two lenses, an objective lens and an eyepiece, much
like the telescope. In the microscope the objective lens forms a real image of the object
that is much larger than the object itself, in much the same way as a projector is used to
project a large image. Then the eyepiece converts the real image into a virtual image and
magnifies the object even more, much like the magnifying glass.
- 88 -
object
objective
lens
eyepiece
lens
viewer
image
Compound lenses
Most optical instruments do not use simple lenses like those diagrammed above. Most
often optical instruments like cameras, projectors and microscopes will use compound
lenses. A compound lens is a lens that is really a series of lenses pressed together.
Compound lenses are used to correct imperfections in the way a simple lens can bring
light into focus. A simple lens generally does not produce a clear focus all the way
across the picture (when the middle is in focus the edges are not), and a simple lens
usually does not focus all colors the same way because the index of refraction is different
for the different colors.
In addition to compound lenses, often lenses are used which actually consist of several
lenses that move with respect to each other. The most common example of such a lens is
a zoom lens. A zoom lens is a camera lens that can change its magnification and stay in
focus all the while.
- 89 -
Your Name _________________
Lab Partner _________________
Experiment 10
Work Sheet
I. Lenses
Set up the light, the slit plate, the parallel ray lens and the ray table as shown.
parallel lens
slit plate
Ray table
cylindrical lens
light
optical bench
lens holder
Turn on the light. Move the parallel ray lens back and forth until the beams of light
coming out of it are parallel (they neither converge nor diverge). When set up like this
the parallel ray lens bends the (diverging) light rays coming from the bulb so that they are
parallel to each other.
slit plate
ray table
light
parallel light beams
parallel ray lens
Discussion: Is the parallel ray lens a converging or diverging lens?
- 90 -
Place the cylindrical lens in the center of the ray table so that the parallel beams go
through the flat face of the lens perpendicular to the flat surface of the lens.
Discussion: Describe what happens to the light rays when they emerge from the round
side of the lens and complete the picture above by tracing the light rays through the lens.
Turn the lens around so that the light rays enter through the round side of the lens.
Discussion: Describe what happens to the light rays when they emerge from the flat side
of the lens and complete the picture above by tracing the light rays through the lens.
Discussion: Does the cylindrical lens act as a converging or diverging lens?
- 91 -
Remove the cylindrical lens. Place the 75 mm convex lens in front of the ray table
(between the parallel ray lens and the ray table).
Discussion: What happens to the light rays when they first leave the 75 mm convex
lens? Complete the picture by tracing the light rays through the lens.
Discussion: Repeat this when the concave lens, and draw a picture showing what
happens.
Discussion: Use a pair of eyeglasses and describe what happens.
Discussion: Of the concave, convex, and eyeglass lenses which are converging and
which are diverging lenses?
Discussion: Was the person owning the eyeglasses near-sighted or far-sighted?
- 92 -
II. The Magnifying Glass and Telescope
Take the 75 mm convex lens and look at one of your fingers through the lens. Move the
lens around until your finger comes into clear focus.
Discussion: Describe the appearance of your finger through the magnifying glass.
Use the 150 mm lens as the objective and the 75 mm lens as the eyepiece of a small
telescope. Hold the lenses in place with your hands. Look at an object across the room
through the telescope. Move one of the lenses back and forth until the object comes into
focus.
Discussion: Describe the appearance of the object through the telescope.
III. Snell’s Law
Place the cylindrical lens on the ray table such that the center of the flat side is at the
center of the ray table, and the flat side is parallel to the 90° line on the table as shown.
Use the slit plate in combination with the
slit mask and the parallel ray lens to
make a single light ray that passes
directly over the center of the ray table as
shown.
90o
component
ni=1
air
incident
light ray
Өr
nr
normal 0o
When the light ray hits the flat face of the
cylindrical lens perpendicular to its
Өr
surface it is not bent by refraction. The
light continues through the lens and out
refracted
through the rounded surface where it is
rotate
light ray
again not bent because it is incident at
0o
90°. However, if you turn the table so
that the light beam is no longer hitting the
flat surface at a 90° angle the light will then be refracted. The refracted beam will
continue through the lens and out through the rounded side where it will not be refracted
because the angle of incidence is still 90°.
0o
- 93 -
Use the protractor scale on the edge of the ray tracing table to measure the refracted
angle, θr, as a function of the angel of incidence, θi; record the results in the table
provided.
Өi
0
10
20
30
40
50
60
70
80
Өr
sinӨi
----
sinӨr
----
sinӨi/sinӨr
----
Avg=
Flat Side
Compute the sine of the refracted angle and record the result in your table. Before you do
this, make sure that your calculator is in degree mode. Check this by punching the
number 90 into your calculator then press the sine button. Your calculator should then
display the number 1. If it does not, you need to change your calculator into degree
mode. After you have calculated the sine of the angles, calculate the ratio of the sinθi to
sinθr and record the result in the first part of the table. This ratio is the index of refraction
for the cylindrical lens. Notice that all the numbers are roughly the same.
90o
Curved Side
Өc
Өc
o
0
ni=1
0o
ni
Өr=90o
rotate
refracted light
ray
0o
nr=1
air
- 94 -
Turn the cylindrical lens around so
that the light beam comes in from
the curved side and goes out
through the flat side. Turn the table
until you find the critical angle. The
critical angle is the incident angle
that causes the refracted angle to hit
90o.
Critical Angle Өc= __________
IV. Conclusion
Write a short paragraph describing what you learned from this experiment.
Was there anything about this experiment that you didn’t understand?
- 95 -
Appendix A-1
Basic Units
Length
Mass
Time
Temperature
Amount of substance
Electric current
Light intensity
Angle
Derived Units
Name
Volume
Density
Velocity (speed)
Angular velocity
Acceleration
Momentum
Angular momentum
Force
Energy (work, heat)
Entropy
Power (work rate)
Frequency
Wavelength
Period
Pressure, stress
Electric charge
Resistance
Electric field
Electric potential (Emf)
Electric flux
Magnetic flux density
Magnetic field intensity
Symbol
l, w, h, d, s, x, y, z
M, m
T
t
N, n
I
IV
Ө, Ф, φ, α, β
Magnetic flux
Φm
Metric Prefixes
Prefix
Symbol
femto
f
pico
p
nano
n
micro
μ
milli
m
centi
c
kilo
k
Mega
M
Giga
G
Tera
T
Peta
P
Name of unit
meter
kilogram
second
Kelvin
mole
Ampere
candela
degree, radian
Unit
m
kg
s
K
mol
A
cd
o
, rad
In terms of base
units
V
D, ρ
v, s
ω
a
p
L
F
E, W, U, K, Q
S
P
f, υ
λ
T
p
q
R
E
V, E
Φe
B (B=μH)
H
Fraction
1/1,000,000,000,000,000
1/1,000,000,000,000
1/1,000,000,000
1/1,000,000
1/1,000
1/100
1,000
1,000,000
1,000,000,000
1,000,000,000,000
1,000,000,000,000,000
Newton
Joule
Watt
Hertz
Pascal
coulomb
ohm
volt
Tesla
Weber
m3
kg/m3
m/s
rad/s
m/s3
kg*m/s
kg*m2/s
N
=
J
=N*m=
J/K
=
W
=J/s=
Hz
=s-1=
m
s
Pa
=N/m2=
C
=
Ω
=V/A=
V/m
=
V
=W/A=
V*m
=
T
=Wb/m2=
A/m
Wb
Decimal
0.000000000000001
0.000000000001
0.000000001
0.000001
0.001
0.01
1,000
1,000,000
1,000,000,000
1,000,000,000,000
1,000,000,000,000,000
- 96 -
=V/s=
kg*m/s2
kg*m2/s2
kg*m2/(s2*K)
kg*m2/s3
1/s
kg/(m*s2)
A*s
kg*m2/(A2*s3)
kg*m/(A*s3)
kg* m2/(A* s3)
kg*m3/(A* s3)
kg/(A* s2)
kg* m2/(A* s2)
Scientific Notation
1 x 10-15
1 x 10-12
1 x 10-09
1 x 10-06
1 x 10-03
1 x 10-02
1 x 10+03
1 x 10+06
1 x 10+09
1 x 10+12
1 x 10+15
Appendix A-2
Fundamental Physical Constants
Atomic mass constant
mu=1/12 m (12C)=1 u=10-3 kg/(mol*NA)
Avogadro's constant
Bohr magneton (eħ/2me)
Bohr radius (4πε0ħ2/mee2)
Boltzmann constant (R/NA)
Classical electron radius (α2a0)
Compton wavelength (h/mec)
Electron g-factor
Electron rest mass
Elementary charge
Faraday's constant
Gravitational constant
Molar gas constant
Molar volume of ideal gas (RT/p)
T=273.15K, p=101.325kPa
Muon rest mass
Nuclear magneton (eħ/2mp)
Neutron rest mass
Pi
Permeability of free space
Permittivity of (1/μ0c2)
Planck constant
Planck mass [(ħc/G)1/2]
Planck length [(ħ/Gc3)1/2]
Planck time[(ħG/c5)1/2]
Proton rest mass
Speed of light in vacuum
Speed of sound (estimate in air)
Standard acceleration of gravity
Standard atmosphere
Stefan-Boltzmann constant [(π2/60)k4/ħ3c2]
Tau rest mass
Earth mass
Earth radius (mean)
Moon mass
Moon radius (mean)
Sun mass
Sun radius (mean)
Earth-sun distance (mean)
Earth-moon distance (mean)
mu
=
1.660538 x 10-27 kg = 9.31494 x 102 MeV/c2
NA
μB
a0
kB
re
λc
ge
me
e
F
G
R
Vm
=
=
=
=
=
=
=
=
=
=
=
=
=
6.022142 x 1023 mol-1
9.27401 x 10-24 J/T
0.529177 x 10-10 m
1.3806503 x 10-23 J/K
2.817940 x 10-15 m
2.426310 x 10-12 m
-2.002319
9.109382 x 10-31 kg = 0.510998 MeV/c2
1.602176 x 10-19C
9.648531 x 104 C/mol
6.673 x 10-11 m3/(kg*s2)
8.314472 J/(mol*K)
2.2414 x 10-2 m3/mol
mμ
μN
mn
π
μ0
ε0
h
mp
lp
tp
mp
c
s
gn
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
1.883531 x 10-28kg = 1.05658 MeV/c2
5.050783 x 10-27J/T
1.674927 x 10-27 kg = 9.39565 MeV/c2
3.1415926536…
4 π x 10-7 N/A2
8.54188 x 10-12 F/m
4.135667 x 10-15 eV*s = 6.626069 MeV/c2
2.1767 x 10-8 kg
1.6160 x 10-35 m
5.3906 x 10-44 s
1.672621 x 10-27 kg = 9.38272 x 10-2 MeV/c2
2.99792458 x 108 m/s
343 m/s
9.80665 m/s2
1.01325 x 105 Pa = 760mm Hg (Torr)
5.670400 x 10-8 W/(m2*K4)
3.16788 x 10-27 kg = 1.77705 x 103 MeV/c2
5.97 x 1024 kg
6.38 x 103 km
7.35 x 1022 kg
1.74 x 103 km
1.99 x 1030 kg
6.96 x 105 km
1.496 x 108 km
3.84 x 105 km
σ
mτ
- 97 -
Appendix B
Significant Digits
Not all numbers are significant.
How to count
Start to the left and go right. Start counting on a number other than zero.
ex. 0.0000001564879503260160 seconds
There are 16 significant digits.
How to record
Direct measurement
When YOU make the measurement
You decide when to stop writing a number down.
ex. Is it 12.3cm or 12.4cm or 12.2cm?
Write 12.3cm ± 0.1cm
You now have 3 significant digits.
*Do not write 12.35cm because it is the 3 that you are unsure of!
*ALWAYS STOP AT THE NUMBER YOU ARE UNSURE OF!
Calculated measurement
-When you have to make a calculation using your data.
Calculators usually give you a lot of digits.
When do you cut it short?
How many digits do you want to save?
When adding or subtracting:
Look at all the numbers, which one has the least decimal place value?
This is where you cut off your answer.
ex. 2.3g+2.45g-6.872g+23.1g=20.978g
1 decimal places
1 decimal places
2 decimal places
3 decimal places
Answer is 21.0g (1 decimal place)
*Keep the zero. It says you know the answer to the tenths place!
- 98 -
When multiplying or dividing:
Count each number’s number of significant digits.
Which one has the least number of digits?
Cut off your answer to that number.
i.e. (2.3m*2.45)/(6.872s*23.1)=.035497583518538m/s
3 digits
2 digits
3 digits
4 digits
Answer is 0.035m/s (2 significant digits).
Uncertainties
All measured values have error involved for different reasons
Direct measurement
When you made the measurement
Calculated measurement
When you make a calculation to get a final answer
Direct Measurement
You decide on the amount of error!
10 cm
length= 20 ± 3cm
width= 10 ± 2cm
height= 10 ± 1cm
10 cm
Depends on:
20
cm
Your ability to measure accurately
Measuring device (did you use a meter stick or a high precision caliper?)
Outside interventions (was there a wind blowing while you were trying to take a
temperature?)
Calculated Measurement
length = 20 ± 3cm, width= 10 ± 2cm, height= 10 ± 1cm
Volume = 20cm*10cm*10cm=2000cm3
Take all the + values and make your calculation (max)
Volume max, Vmax= (23cm*12cm*11cm)=3036cm3
Take all the – values and make your calculation (min)
Volume min, Vmin= (17cm*8cm*9cm)=1224cm3
The uncertainty in the volume,
V − Vmin
∆V = max
2
∆V= (3036cm3 - 1224cm3)/2=906cm3



Complete answer V ± ∆V=2000 ± 906cm3
- 99 -
This is done for all types of calculations, not only for volume. If we are doing an area,
you would use A=length*width. If we are finding densities, you would use
Density=mass/volume.
Scientific Notation
Often a number is vague in its number of significant digits. Scientific notation is used to
clarify and to simplify numbers.
A number has only 2 significant digits but the number is 49276400000, there is a problem
since 49000000000 has 11 significant digits. How can you reduce this to 2 significant
digits without changing the number? What if the number was 0.00000000049? It has the
proper number of significant digits but can be simplified.
1. Move the decimal place until it is before the first digit.
ex1: 4.9276400000
ex2: 0.00000000049
x
2. Multiply by 10 , where x is the number of times you had to move the decimal
from its original position.
ex1: 4.9276400000 x 1010
ex2: 00000000004.9
3. Remember that moving the decimal to the left is 10+x, moving to the right is 10-x.
4. Reduce to the proper number of significant digits.
ex1: 4.9 x 1010
ex2: 4.9 x 10-10
- 100 -
- 101 -
- 102 -