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Physics 2A
Lab Exercise
Manual
version 1.0
by Stephanie Dickson and David Newton
Copyright © 2000
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Lab Book Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Uncertainties and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Including error bars on a graph and how to use them . . . . . . . . . . . . . . . 9
Significant Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1. Measurement and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Measuring Your Reaction Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4. Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5. Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6. The Atwood's Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7. The Ballistic Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8. The Slingshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
9. Centripetal Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
10. Mass on a Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
11. The Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3
Introduction
Attendance: Laboratory attendance is mandatory. To miss a lab is to miss those skills learned only by doing. A lab
cannot be made up outside of class or during another week since the equipment is then put away. Check with your
instructor for a specific attendance policy; some teachers will drop a student from the course after having missed
only one or two labs regardless of the validity of the absence.
W hat you must bring to class: Lab book (then kept in the classroom until after the end of the quarter), ruler, pen,
scientific calculator and this lab exercise manual.
Lab book policy: A quadrille ruled, non-spiral bound standard lab book will do. Put your name and section number
on the outside front cover. Lab books will be kept in class and not taken home, so your work must be completed in
class. Save the first two pages for a table of contents. Number the rest of the pages of the lab book in the top right
corner for your table of contents. If you use both sides of a sheet then you should number both sides. Do this as soon
as you buy your book, not as the lab progresses through the quarter. Then complete your table of contents as the lab
progresses.
You must write in your lab book with pen only. This eliminates the temptation to erase what you have
written because you think you have made a mistake. Do not erase anything you have written in your lab book. Do
not use White Out! If you think you've made a mistake, cross out what you have just written (so you can still read it)
and re-write the correct version near it. The reason you don't want to erase anything in your book and that mistakes
should be crossed out so you can still read them is that occasionally what you thought was a mistake may turn out to
be correct and you don't want to loose the original information. An objection might be raised that doing so renders
the lab book too messy. A "messy" lab book is not a bad lab book. Of course, you must be able to read and locate
data in your book, but a pristine lab book is usually contrived and artificial. Real lab books are not perfect and
beautiful records of work. They are written logs of what you have done while you are doing it! As such they may
not always be error free. Do not interpret these remarks to mean a sloppy book is desirable or even tolerable. Your
book must be a clear, chronological "diary" of tasks performed in the lab.
Absolutely no scratch paper is allowed. That includes any loose papers in your lab book. Loose or
scratch paper of any kind will be thrown away by your instructor as soon as they are spotted and any work done on
these will be lost. This rule will be strictly enforced! This ensures all original data, calculations and ideas are
entered chronologically and nothing is lost. The mentality of taking the raw original data on a scrap paper and later
transfering it to the lab book neatness is wrong since it allows for mistakes to be made during data transfer.
W hen a graph is drawn, try plotting each point as soon as it is measured. This enables you to spot trends
and mistakes quickly without wasting time. This may mean that your graph is not as good as if you had more time to
work on it -- that's okay! Quality graphs are a different matter and for the most part a quality graph is best drawn by
a computer. Hopefully some computer analysis will be done in class, but the emphasis will not be on using
computers in this course. All graphs in the lab book are full page graphs. A full page graph is defined as follows:
draw a rectangle inscribing all the data points. To qualify as a full page graph, the rectangle must cover at least half
the page in both dimensions.
4
Lab Book Structure
Your lab book should have a discernable organization to it. The exact structure depends on the kind of
experiment you are performing. But all good lab books have one thing in common and that is the ability to yield
information upon examination. Being able to retrieve information from a lab book is the mark of a successful book.
Anyone should be able to peruse your book and from the book only understand what you have done.
In beginning an experiment, you may be eager to start taking your data immediately. Resist this temptation,
as it leads to disorganized thinking and a disorganized lab book. You should first organize your thoughts about the
structure of your experiment in your book. Design your experiment before you take your measurements, but remain
flexible because as your experiment progresses, you might have to alter your original intentions.
Your section headings should stand out from any surrounding text. Below are some standard sections that
are found in virtually all lab books. Use what fits your needs and consult your instructor as necessary.
Equipment List:
Record a list of the equipment used. W rite down the i.d. number found on the equipment if available so you can get
the same equipment if you need to repeat the experiment.
Purpose or Objective:
Any experiment tries to accomplish something. State your goal clearly. Proclaim your expectations if you have any.
Theory:
Derive pertinent equations and explain relevant concepts. Use diagrams as much as possible. Develop the basis for
your uncertainty analysis here.
Diagrams:
Lucid diagrams are invaluable as an aid to explanations. Drawings of equipment, schematic circuit diagrams, or any
other relevant subject are encouraged.
Procedure:
Discuss your plan for performing the experiment. Don't describe the content of your entire lab. W rite down your plan
for "running" your experiment and taking measurements in a step-wise, linear fashion. It is important that this section
be described in detail as it is in your procedure section that you or anyone else would re-create your experiment as
you performed it. If you decide to change your procedure as the lab progresses, make sure the change is labeled as a
procedure and that you go back in your book to the original procedure section and clearly label page references to
your change. Draw labeled diagrams to illustrate how the equipment is set up and how it is to be used.
Data:
Record all your measurements in a clear and readable fashion. Any number measured in the lab must go directly in
the lab book. Use "table" form for legibility. Record the measurement with the appropriate units and uncertainty. Do
not copy from your partner’s lab book. Each person must write the raw data in their own book directly from the
measurement.
Analysis: Together with your uncertainty analysis, compare your measured and calculated results to your numerical
5
expectations. Graphs are helpful. Every lab book will have an analysis using one or more of the following::
1. Calculations: Show one "sample calculation" in complete detail, then write the results only of other
similar calculations in a table form. Do your uncertainty calculations here.
2. Results: Show all your final relevant numerical results. Boxing important values will allow anyone quick
access to your final important calculations.
3. Graphs: If possible, draw your graphs while the data is being taken, point by point, so measuring errors
and blunders can be easily detected before you end your data measurements. For better quality graphs drawn after
the experiment, use a computer if available. All hand draw graphs should occupy at least half of one full page. Draw
your data points with a sharp pencil. Never "connect the dots" on your graph one point to the next! Draw an "eyeball" best fit line through the data points using a ruler if a linear relationship is expected. Understand that the "best"
line through your data may not be through even one data point.
Conclusion or Summary:
Discuss the significance of your results. Sometimes an idea that comes to you spontaneously in the lab is only
recalled if jotted down somewhere in your lab book. Such an idea would be discussed in more detail in this section.
Use complete sentences for clarity. Avoid discussing "human error" unless outright blunders were made. Use
numbers with uncertainties. Discuss needed improvements in technique, procedure, and equipment. Discuss methods
to eliminate or minimize systematic errors.
6
Experimental Technique
After drawing a diagram of your setup and outlining your procedure, you are ready to begin the experiment.
Do a few dry runs to understand how the experiment goes. The goal of the experimenter is to conduct the
experiment in the most honest way. In some experiments, your result, with careful, thoughtful data taking, will agree
with an accepted value within uncertainty. In some, it will not. Do not look to the outcome during the data taking
phase of your experiment. Instead, record all measurements into your lab book without trying to second guess what
those measurements “should be.” Here are some hints to help you get the best results for your experiment:
1. Try to understand how the equipment works.
a. Whenever you look at a new piece of equipment, find the zero point. For example, what does
the vernier caliper look like when it reads zero? W hat does the triple beam look like when it reads
zero? Some equipment might need to be “zeroed,” or adjusted so that when you are measuring
zero, it reads zero.
b. Figure out what each division means. On the large timers, for example, one sweep of the large
hand equals 10 seconds. That means each tick of the small hand is also equal to 10 seconds and so
on.
2. Try to eliminate equipment reading errors.
a. When using a meter stick, put the meter stick on edge to avoid parallax error.
b. Never use the edge of the meter stick to mark the zero point of your data taking. Often, the
edges are not cut precisely or worn with time. Instead, set the meter stick down randomly and
begin your measurement wherever it lands.
c. W henever possible, verify equipment readings with calibrated sources. For example, you might
weigh a known mass on the triple beam. Or you might use the second hand on your wristwatch to
check the timer.
Use of Laboratory Equipment:
Leave all your equipment set up until your experiment is completely finished; if a mistake has been made
and you need to retake some data or check your apparatus, you would rather not have to set up your entire lab all
over again. A very important, and often overlooked aspect of proper laboratory procedure is careful and considerate
use of laboratory equipment. Some of the equipment is easily broken and not easily replaced. Take time to ensure
that you know how to use the equipment, listen to the hints from your instructor about which parts are easily broken,
do not overload the equipment. All equipment must be carefully and neatly replaced on the lab cart. Equipment left
in the lab can be lost or stolen.
7
Uncertainties and Statistics
Any scientific number, having been obtained by measurement or calculation, never has an exact, single value.
Even when written as just a single number (like g = 9.8 m/s 2), it is understood an interval of possible values. This interval
of values is known as the most probable range (M PR) of the scientific measurement.
There are two basic ways of writing a scientific value that clearly states the MPR of the value.
As an example of the first method, say the recorded measurement of a length is 3.5 cm. This statement is not
saying the length is exactly 3.5 cm, but that the length lies somewhere between 3.45 and 3.55 cm. So even though only one
number was stated, an MPR is implied. Under these conditions we say the M PR of a value stated as just one number is
within plus and minus one half the unit of the last decimal place stated. This is the method commonly used in texts
where a measured value is stated as one number.
As another example, in Giancoli's text (5 th edtion, page 34), the accepted value for the gravity field at the surface
of the earth is g = 9.80 m/s 2 . This value is ultimately obtained from measurements and as such is not an exact value. See if
you can apply the above information and write down the MPR for this stated value of the gravity field. The correct answer
is given as a footnote at the bottom of this page, but don't look until after you've tried to get the right answer.1
To write the above length measurement (3.5 cm) with an explicit most probable range we could just write out the
range {e.g. M PR = (3.45 - 3.55) cm}, or as is more commonly done, we could write the value as a best or most probable
value (M PV) plus or minus what is called the "absolute uncertainty" in the MPV, *MPV. This method is the second way of
stating the MPR of a measurement and is preferred. In the above example the format for this method would be as follow:
(lbest ± *l) = (3.50 ± 0.05) cm
See that the MPV is not written as 3.5 but as 3.50 so that the decimal places agree between the MPV and the
absolute uncertainty (both are to the hundredths of a centimeter or two decimal places). That the decimal places agree
between the MPV and the absolute uncertainty is most important; do not neglect this point.
1
The MPR for 9.80 m/s2 would be (9.795 to 9.805) m/s2 or (9.800±0.005)m/s2 .
8
How to find the M PV and the M PR statistically:
If a measurement can be repeated and is expected to yield the same result each time, then a statistical method for
associating a most probable range with the measurement is used. The following method is valid only if at least five
repeated measurements can be taken.
From a given set of data values x i, the best or most probable value is simply the average value, &
x . The absolute
uncertainty, the uncertainty in the mean value, is the so-called "standard deviation of the mean" or the "standard error".
The arithmetical formulas for the values are given below:
The formulas stated above are in terms of the usual statistical symbols. To translate these symbols into the ones used in
uncertainty analysis follow this guide:
Example: Say five values have been measured.From using the formulas above calculate the mean and standard deviation
of the mean of this data set and confirm that its value is : (4.36 ± 0.05)cm. See that the MPR associated with this value is:
4.31 (4.36-0.05) to 4.41 (4.36+0.05)
Data Table:
Xi
(cm)
X1
4.4
X2
4.3
X3
4.4
X4
4.2
X5
4.5
Note that although all the measurements were taken to only one decimal place (e.g., 4.4,
4.3,etc..), by using the statistical method two decimal places for our best value were
obtained (4.36). W e see that by repeating the measurement of a value that should be the
same, the more times we repeat the measurement the greater the precision of our result.
Remember, the uncertainty in &
X is not the standard
deviation but the standard deviation of the mean. It is also conventional to write the
uncertainty in the most probable value to one significant figure only. Having done this,
the place to where the absolute uncertainty is rounded to one sig fig determines the place
to where the MPV is rounded. This last point most important. In the above example, you
might have calculated the standard deviation of the mean to be 0.05099 cm. However you
will always round this value to one significant figure- that is, 0.05. See that the
uncertainty has the same units as the measured value.
9
Including error bars on a graph and how to use them
Graphs involving measured values typically include "error bars". The
error bars graphically represent the absolute uncertainty of a measurement and
allow the calculation of the uncertainty in the slope's measurement.
In the diagram shown to the right, the data point is plotted in the usual
fashion; as shown, the absolute uncertainty is drawn above the data point and then
below the data point representing the "plus and minus" aspect of the absolute
uncertainty (i.e., measurement = value ± absolute uncertainty).
An error bar graphically expresses the absolute
uncertainty of the measurement. The
measurement is (1.2 ± 0.1)cm
.
After all the data points with error bars are drawn,
the "best fit" line is drawn as usual. W ith error bars, the new
feature is that two more lines can also be drawn that
represent the maximum possible slope and the minimum
possible slope as shown in the diagram to the right. This
allows the calculation of the absolute uncertainty in the
slope. Admittedly, this is a subjective process; a more
analytical process exists but will not be discussed here. As
an example of this technique, say the graph on the right has
a measured slope of the best fit line of 3.4 cm/s (note that
the slope has the units of rise/run). If the maximum slope
line has a measured slope value of 3.9 cm/s and the
minimum slope line has a slope of 2.9 cm/s, then the final
slope value may be written with an absolute uncertainty as
(3.4 ± 0.5) cm/s.
The two dotted lines show a maximum and minimum slope that can be used to
construct the uncertainty in the slope. The solid line is the best fit line.
For another example, say that from theory it is
known that the slope m of a graph is related to a quantity to
be found Q by the equation: Q = 2.5 m. If the slope is known with an absolute uncertainty as m = 23.5 ± 0.8 then Q with
an absolute uncertainty would be equal to 59 ± 2. [EXPLANATION: Q = (2.5)(23.5) = 58.75 : the uncertainty in Q is
(2.5)(0.8) = 2.0 or just 2 to one significant figure (where 2 is rounded to the units place), so 58.75 is also rounded to the
units place.]
10
Significant Figures
A significant figure is a digit in a number representing a quantity, other than a leading or trailing zero introduced only to
mark the decimal place position. The meaningful digit in a number furthest to the right is called the least significant digit.
One use of significant figures allows a quick and casual approximation of the final value of calculations. Also,
looking at the amount of significant figures in a number gives an indication of the precision of that value. As an example,
if you divide 2 by 3, your calculator will display 0.666666667 or as many sixes as the display will allow. It seems
unreasonable that values containing only one significant digit (2 and 3) could yield a calculated answer nine digits long.
This example violates our intuitive notion of how measured or known values yield final values in calculations. You can't
get more precision than what you started with.
Use the following rule in writing final answers involving division or multiplication:
A final calculated value has the same number of significant figures as the least amount of significant figures found
in any one factor of the calculation.
In the above example, since there is only one "sig fig" in each initial value, the final value cannot have more than
one sig fig. The correct answer is: 2 ÷ 3 = 0.7. The zero in front of the decimal point is used to mark the decimal place
(this is considered good form); this zero is not a significant digit.
In addition and subtraction, use the significant figures to the right of the decimal place to guide your final answer.
Example: 234.45 + 52.432 = 286.88.
Here, 234.45 has only two digits to the right of the decimal place so the answer will have two digits to the right of
the decimal place as well.
W hen using transcendental functions, the significant figure found in the argument of the function is equal to the
significant figures stated in the result.
Example: ln(11.24) = 2.419; four sig figs in each.
Sometimes it is impossible to tell how many significant figures are stated in a vaguely written value.
Example: How many significant figures are in this value: 2100?
The answer is ambiguous. There are three possible answers: two, three, or four. The two zeroes after the 21 make
the answer unclear. If the two zeroes are both not significant, they serve only to mark the position of the decimal place. In
that case the answer would be: 2100 is two sig figs. But perhaps the zero right after the 21 is significant but the next zero
is not. Then the correct answer would be three significant figures. Finally both zeroes may be significant in which case the
correct answer would be four significant figures. In this last case some people would say "2100." is the correct way of
writing this value if both zeroes were significant. This convention is fine, but is not universally followed so be careful.
W e have just seen that the existence of zeroes in a number may confuse the issue of how many sig figs the
number contains. To avoid this ambiguity, use "scientific notation".
11
Stating numbers in scientific notation eliminates all ambiguities with regard to how many significant figures a
number has.
In the previous example, 2.1 x 10 3 is a number with two sig figs, 2.10 x 10 3 has three sig figs, and 2.100 x 10 3 has
four significant figures. W ritten in this fashion there is no problem determining exactly how many significant figures are
stated in any value.
Another method besides scientific notation can be used to avoid ambiguities in stating a result. Place a line
above or below the least significant digit. As an example 24,0 &
0 0 has four significant figures.
These methods should be used in the solution of homework problems. In formal lab work however, these "rules
of thumb" for significant figures are not accurate enough. Instead, the significant figures of a final calculated result are
determined by statistics and uncertainty propagation. The details of these methods are beyond the scope of this manual.
12
Here are some more examples. In one or two complete sentences, you should be able to explain how the
significant figures in each have been obtained.
13.0010
-- six sig figs.
0.006
-- one sig fig.
0.0060
-- two sig figs.
1402
-- four sig figs.
1200.00
-- six sig figs.
1200
-- ambiguous; could be two, three, or four sig figs.
1.20 x 10 4 -- three sig figs.
100.
-- three sig figs.
Arithmetical examples:
11.21 x 1.23 = 13.8
Multiplication: 13.8 has three sig figs; this matches the fewest amount of sig figs used in the calculation: 1.23
(0.324)(2.2)/3.0012 = 0.24
Multiplication and division: 2.2 has the least amount of sig figs (two sig figs), so the answer has two sig figs.
8.1 + 9.3 - 4.5 + 6.5 = 19.4
All numbers have one significant digit to the right of the decimal place so the answer has one significant digit to the right
of the decimal; note the answer has three sig figs while all the other numbers only have two.
13
1. Measurement and Uncertainty
Equipment list:
meter stick
Purpose: To understand and apply the techniques associated with uncertainties in scientific measurements and
calculations.
Introduction: Measurements always have an uncertainty associated with them. The association of an uncertainty with a
measurement may be ignored under casual or special circumstances, but any precise scientific precedure involves at least
the recording and justification of an uncertainty when a measurement is taken. W henever a measurement is read from a
scale (as opposed to a digital readout) a "rule of thumb" for the uncertainty associated with the measurement is that the
uncertainty is equal to the "least count" of the scale used. The least count of a scale is equal to the smallest increment of
measure on that scale. For example on the meter stick provided in this lab, the smallest increment on that scale is the
millimeter or 0.1 centimeters. Thus any measurement made with the meter stick is uncertain by one tenth of a centimeter.
Another way of expressing this uncertainty is by saying a measurement read from the meter stick lies within a range of
"most probable values" (MPR). For example, say the length of an object is measured to be 13.46 cm. (note that the final
decimal place is an "interpolated" value subject to variation in repeated readings). Since the uncertainty of the reading is a
tenth of a centimeter we write the measured value with its uncertainty as (13.46 ± 0.05)cm. It is most important to see that
the decimal places in the measurement and the uncertainty are the same (here to the "hundredths" place). Also, see that the
plus and minus of 0.05 represents a total uncertainty of 0.1 cm (0.05 + 0.05 = 0.10). This last point may need some
reflection but is most important.
As an alternative to the plus and minus (±) notation we could also and perhaps more clearly write the measurement as the
most probable range style mentioned previously. The measurement then would be written in the MPR format as 13.41 to
13.51 cm. That is, we say the most probable value of the length of the measured object lies between 14.41 and 13.51
centimeters. Notice that this range is equal to a tenth of a centimeter which of course is the uncertainty in the reading.
Other more sophisticated methods exist to determine uncertainties but this method will suffice for the now.
Procedure: Using the meter stick provided, record the length and width of the table top in your lab. For each
measurement estimate a "scale uncertainty". Calculate the area of the table top. Then for your calculation of the area
include an expression of its uncertainty also. Show all calculations in your lab book.
Results: Express the area of your table as a most probable range (e.g. 134.45 - 145.48 square centimeters). Significant
Figures: Determining the correct number of significant figures for the final answer is important and as above, there are
other methods that can be applied. However for this lab, when calculating the area of the table the significant figure of the
final answer will simply be equal to the least number of significant figures in either of the two products.
Conclusion: Discuss methods of improvements such that future attempts could avoid the mistakes you made.
14
2. Measuring Your Reaction Time
Equipment List:
meter stick
Goal: To apply kinematics to determine your reaction time and to analyze your results with some basic techniques.
Introduction:
A falling object speeds up uniformly, that is, the object has a constant acceleration. If we neglect air resistance,
the acceleration of a falling object is about 9.8 m/s2 and does not depend on the object's weight.
If an object falls from rest (i.e., its initial velocity is zero) then the distance the object falls, )y, is related to the
time it has fallen, t, by the equation:
)y = ½gt2 (where g is 9.8 m/s 2.)
The idea of this lab is borrowed from an old trick. One person holds a dollar bill at its top end lengthwise and the
other person, with two fingers held close to the bottom of the bill, tries to grab the bill after the first person lets go. The
person trying to grab the bill does not know when the person holding it will let go. Surprisingly, often the person cannot
catch the bill since it speeds up so quickly under the acceleration of gravity that it has fallen away from the fingers of the
"grabber" before the fingers have closed. The quicker your hand-eye reaction time, the better your chances for grabbing
the dollar bill.
Procedure:
First, use the meter stick instead of the dollar bill. Together with your lab partner, devise a method to measure
your reaction time; clearly denote the method in your lab book under procedure. As a suggestion, your lab partner could
hold the meter stick and let go at a time you cannot predict. The time from when the stick is let go to when it is caught is
your reaction time. A timing clock and a meter stick are provided; how you use either or both of these devices is your
decision not your instructor's. There may be more than one way to measure your reaction time; you need only use one
method.
Take many trials (at least twenty) and then switch with your partner. As you "run" your experiment, draw a graph
of your reaction time as a function of the number of the trial. Draw this graph as you take your data - point by point.
See if you can spot a trend.
Questions:
1. Can you improve your reaction time with practice? If so, by what percent?
2. W hat is your average reaction time?
3. Are you actually measuring two of your reaction times or just one? Are you measuring your lab partner's reaction time
as well as your own?
Follow up:
Knowing your average reaction time, take out a dollar bill (or a five or just a piece of paper cut to the same
length as a dollar bill) and measure its length. Predict on the basis of this measurement and the knowledge of your own
reaction time whether you should be able to catch a dollar bill when your partner lets go of it. After writing down your
prediction, try catching the bill with your partner. W hat differences (if any) are there between using the meter stick and
using the dollar bill? Switch roles with your lab partner and repeat the experiment and compare your two reaction times.
15
3. Density
Equipment list:
one "squared" aluminum block
vernier caliper
triple beam mass balance (Pan Balance)
Purpose: To gain confidence in measurements and uncertainties.
Theory: About density. All matter has mass. Any object has its mass contained in a definable volume. Density (D) is the
measure of how much mass is contained in an object's volume:
Density = M ass/Volume
Procedure: Measure the length, height, width, and mass of your block. Each measurement will have an uncertainty;
record the uncertainty in your lab book with the measurement. Justify the uncertainty's value.
Calculations:
A. Calculate the volume of your block from your length, height and width values using no uncertainties. Then using your
uncertainties, calculate the V min and V max values and express the volume as a most probable range (MPR). Use the units of
cm 3.
B. Calculate the density of your block to four significant figures without uncertainties. That is, divide the mass of the
block by the volume you calculated in part A and write down the density of the block with no uncertainties. Then calculate
the D min and D max values using your MPR values for the volume and the MPR values for your mass measurement. As an
example, say your mass measurement was M ± *m = (12 ± 2)kg and your V min = 3.2 m 3 and V max = 4.3 m 3. Then you should
verify that:
D min = M min/V max = 2.3 kg/m 3 and D max = M max/V min = 4.4 kg/m 3
Analysis: Compare your calculated value for the density with the known value in the units g/cm 3. 2 Perform a "discrepancy
test" 3 (i.e. calculate percent error). Your result should be less than one percent from the accepted value. If your result is
greater than four or five percent, it is unacceptable and you might be making a mistake.
Conclusion: Using complete sentences (no equations), state your results and discuss possible methods of improvement.
Does your discrepancy test yield a tolerable value? Does the accepted value lie within the most probable range of the
calculated value?
2
3
Known value: DA l = 2.699 g/cm3 at room temperature.
A discrepancy test is a comparison between a calculated value and a known value. It is useful when the two numbers are supposed
to be, at least in principle, the same:
[(known-calculated)/known] X 100%
16
4. Projectile Motion
Equipment list:
sliding ramp, ball, and stand
meter stick
”detecting” paper (to record the ball’s landing positions)
masking tape (to hold the paper down on the lab bench)
plumb bob (for alignments as necessary)
Purpose: To explore the properties of projectile motion.
Introduction: The idea of this lab is to
launch a projectile (the steel ball) with
the same horizontal launch velocity
V ox but at a variety of elevations. Of
course, the greater the elevation of the
launch, the farther the ball will travel
horizontally if the launch speed, V 0x, is
always the same. It is important that
the initial launch velocity of the ball
should be horizontal only; this may be
difficult to achieve given the somewhat
sloppy nature of the equipment. You
will change the elevation, h, of the
launch ramp above the lab table (not
the height of the ball falling along the
ramp which should be kept constant
ensuring a constant launch speed) and
measure the subsequent horizontal
distance, R, the ball travels just after
leaving the ramp.
By drawing a graph of pairs of
points: (R i, h i½) , you can determine
the horizontal launch speed V ox..
Theory: Using kinematics, derive an equation that relates R i to h i½ in terms of g and V ox. You should have an equation that
is linear in h ½ and R. As a hint in doing this, find the time of flight of the ball first (the vertical parameters determine the
time of flight: V 0y = 0, h and g). After you have found the equation for the time of flight, apply it to the horizontal velocity.
Recall that in projectile motion the horizontal velocity is never accelerates. Relate the range R to V 0x and the time of flight
equation to get one final equation in terms of R, V 0x, h, and g only.
Procedure: First use masking tape to close any holes on your lab table; the balls can fall down those holes and are
extremely difficult to retrieve! Set up the apparatus as shown and tape down the ditto master where the ball will land on
the table. W hen the ball lands on the ditto master it will make a mark allowing you to measure R. For each given height h,
perform three trials for three values of R. You will use the average of the three values of R when plotting your data.
Record the height h for each given range R.
17
Adjust the launch height to obtain five different pairs of data (R i, h i). Choose your five different heights h to span most of
the available height. Remember for each trial the vertical distance the ball falls along the launch ramp should be a constant
and the ball should be launched horizontally.
Analysis: For each value of h i also calculate h i½, the square root of h i. Draw a full page graph of h i½ on the vertical axis
and R ave-i on the x axis. Convince yourself that doing so will result in the graph of a straight line. A full page graph means
that the scaling of each axis should have the data points spread over most of the page. Compare your equation derived in
the theory section to the classic y = mx + b equation of a straight line. Relate each part of your derived equation to the
slope, y, and x values. From your derived equation take the variables that determine the slope and set them equal to the
slope you measured on your graph m. You now should have an equation in terms of slope m, g, and V 0x (note: h and R
have now been eliminated from any further calculations; they are only used to draw your graph, not to directly calculate
the launch speed). Solve this equation for V 0x and calculate its value. Remember g = 980 cm/s 2 when h is measured in
centimeters. W hat is the value of the y-intercept b?
After you have plotted your data points draw a "best fit eyeball" straight line through the points with your ruler.
The best fit line through a set of points does not necessarily intersect any of the data points. Do not include the origin as a
data point and don’t use the origin to draw your best line; only measured data should be used to draw your best line. After
you have drawn your line then measure the slope. Show your measurement of the slope on the graph itself. Choose two
points on your line to determine the slope that are as far apart as possible; they should not be data points. From the
measurement of the slope of the straight line you should be able to calculate V ox.
Conclusion: Do you have any reason to believe that your answer for V ox is consistently too high or too low? This would
be called a systematic error in your experiment. How might you eliminate this uncertainty?
18
5. Friction
Equipment list:
friction blocks
blue spring scales
large weight set
pan or digital balance
Purpose:
To explore the properties of friction and calculate a kinetic and static coefficient of friction between a wooden block and
the surface below it.
Introduction:
As relevant to this lab, the following three statements, labeled as claims, are part of the lore of friction.
Claim 1. The coefficient of friction is independent of the weight of the block and the force of friction is linearly
proportional to the weight of the block.
Mathematically this statement is expressed as follows:
Ff = : FN
Claim 2. The force of friction depends on the type of surfaces interacting but is independent of the area of contact
between them.
If this statement is correct you should measure the same force of friction regardless of which of the block's six
possible surfaces is being dragged across the table.
Claim 3. The coefficient of static friction is typically greater than the coefficient of kinetic friction ( :s > : k).
Measure :s and compare to your value of :k as measured earlier. If you have uncertainties, you can make a
scientific claim as to which coefficient is greater.
Preparation:
This lab investigates all three of the above statements to one degree of thoroughness or another. But you will
need to know how to take measurements with the spring scale and then how to relate those values to the force of friction.
1. Starting with Newton’s second law, derive a formula that relates the coefficient of kinetic friction, :k, to the horizontal
tension force, F T (measured by the spring scale) when the block moves at constant velocity. This allows you to know the
friction force by taking a measurement from the spring scale.
2. Taking measurements with the spring scale. The spring scale has two sets of graduations on it. One set reads mass in
grams and the other set reads force in newtons. M ake sure the spring scale measurements for the tension force are
taken in new tons, not grams. Be careful turning the white plastic nut found at the top of the scale, it may come off the
scale housing and then the whole thing will fall apart; its not as easy to put back together as it is to take apart!
3. Determining the uncertainty of the spring scale. Using a "rule of thumb" method for the absolute uncertainty of a scale
reading (i.e., the absolute uncertainty is equal to half the smallest increment on the scale), record in your lab book the
absolute uncertainty associated with a measurement from the scale.
19
Procedure:
Testing claim 1. This is the main part of the lab and requires a full experimental write up and a full page graph. The goal
is to calculate the kinetic coefficient of friction between your block and the table by measuring the slope of a graph plotted
from experimental data. You may choose to use a paper surface under the block rather than the bench surface directly;
often the bench surface is too irregular for consistent results. See your instructor for the availability of the paper. The
relationship between claim 1 stated at the beginning of this write up and finding :k from a linear graph is left for the reader
to ponder.
Pull the block at constant velocity with the spring scale. From the spring scale reading, find the force of friction
on the block. By adding weights to the top of the block and thereby increasing the normal force on the block, a new force
of friction can be measured with the spring scale by again pulling the newly weighted block at constant velocity. Take at
least five data pairs of the friction force (as ultimately measured by the spring scale) and the normal force on the block
(include the mass of the block and all weights added to the top of the block. From these data construct a graph whose
slope is equal to the coefficient of kinetic friction. Measure the slope and find the coefficient.
For greater depth, find the uncertainties in the friction force and put error bars on your data points; Do a
minimum and maximum slope analysis to find the uncertainty in the coefficient of kinetic friction (i.e., the slope of your
graph).
Testing claim 2. This part can be done casually without a formal experimental process. Pull the block at constant velocity
with as many different sides in contact with the surface as is possible (at least two different sides). Does the force of
friction change (within the uncertainty of your measurements) when the block is on different sides?
Testing claim 3. After the coefficient of kinetic friction has been found, from claim 1 above, calculate the coefficient of
static friction. This will be done casually (like the testing of claim 2) with a few measurements and then a direct
calculation. W ith the block at rest, gently pull the block with the spring scale. The spring scale’s reading (and so the force
of friction) will increase to its maximum value where the block will break free and start to slip. The reading on the spring
scale when the block just breaks free and slips is the maximum force of static friction on the block. From this and knowing
the mass of the block, the coefficient of static friction may be directly calculated.
CONCLUSION:
On the basis of your results, refute or confirm the above three claims. Defend your results.
TIPS:
C
C
C
C
W hile pulling the block, the spring scale should be held in a horizontal position so that the scale neither pulls up
or down on the block.
Make sure the spring scale reads zero when all the slack is taken up in the spring scale mechanism as you first
start to pull on it.
It is okay (actually, recommended) to hold the spring scale by its housing while pulling the block.
Make sure no dust, grime, or errant matter is between the block and the table.
20
6. The Atwood's Machine
Equipment list:
Atwood's apparatus
stop watch
meter stick
pan or digital balance
full weight set and hangers
Purpose: To conduct an experiment involving Newton's second law.
Theory: You should derive an equation that yields a theoretical
acceleration, a t, of the Atwood's Machine as well as the tension in the string of
the Atwood's Machine using Newton's Laws of motion. For your measurements,
you should use kinematics to derive an equation that yields a calculated
acceleration a c in terms of measured values, time t, displacement )x, and initial
speed v i. See your instructor if necessary.
Procedure:
1. Set up the experiment as shown.
2. Choose two masses m 1 and m 2 that differ by about twenty grams.
3. Having suspended the masses and holding them still, let them go and measure
a time and distance so you can calculate the resulting acceleration of the masses.
4. Practice measuring the time and distance values with your lab partner.
5. Repeat your procedure for a second pair of masses, m 1 and m 2, that differ by
about ten grams. For each pair of masses you should do five different "runs" so
you can calculate an average acceleration.
Data: Construct a data table showing, for each pair of masses, your five
different x and t values.
Calculations: From your data calculate the accelerations, a c, for each pair of
masses. Also calculate a theoretical acceleration, a t, for each m 1 and m 2 pair.
Results: Compare your theoretical and calculated acceleration for each pair of
masses.
Conclusion: Suggest reason for the difference between your theoretical and
calculated accelerations. W ould there be a systematic error responsible for this
difference? If so, what?
21
7. The Ballistic Pendulum
Purpose: To predict the landing point of a projectile based on an experiment with a ballistic pendulum and then see if it
actually lands at that point.
Equipment:
ballistic pendulum apparatus
magic “detecting paper” (as was used in the projectile motion lab)
masking tape
meter stick and ruler
pan or digital balance
Theory:
W ith a ballistic pendulum, a ball
is fired horizontally into a pendulum
catcher; together, the ball and catcher
swing up to a stop through a vertical
height, h. The analysis of this event
occurs in two parts; the first part uses
momentum conservation and the second
part uses energy conservation.
For the first part, the ball and
pendulum have an inelastic collision, they
stick together. For the collision between
the ball and catcher system, momentum
is conserved. The initial firing speed of
the ball can then be related to the speed
of the ball and catcher when stuck
together just after the collision.
After the collision, the ball and
pendulum rise together. Their initial
kinetic energy just after the collision is
turned into gravitational potential energy
after swinging up to a stop a vertical
distance, h.. Use conservation of energy
to relate the speed of the ball and catcher
just after the collision to the vertical
height h.
One can then relate the initial firing speed of the ball before it collides with the catcher to the height h which can
be measured. After finding the initial firing speed of the ball, using kinematics, find the range R the ball would travel after
being fired horizontally a distance d above the floor. Since you can measure the distance d (it is the height from the
bottom of the ball as it just leaves the firing mechanism down to the floor where it lands) and you know the firing speed of
the ball, you can predict where the ball will land.
The final equation used to calculate R should be in terms of what you have measured: the mass of the ball, the
mass of the catcher, the height h, and the height d. Note: actually calculating the firing speed of the ball as a number is
irrelevant and need not be done.
22
Procedure: 1. Set up the apparatus. Acquaint yourself with its function and alignments.
2. Find the center of mass of the ball and pendulum when they are connected; it is a mark etched into the
catcher. Consult your instructor if you can’t find it. The height h to be measured is the vertical distance
the center of mass displaces in swinging up to a stop.
3. Measure the mass of the ball. Note the mass of the catcher/pendulum, it is also etched somewhere.
4. Proceed to fire the ball into the pendulum/catcher, measure h. (Remember, this is the position of the
center of mass)
W ARNING: TAKE CARE W HEN FIRING THE BALL. SOM EONE COULD BE SERIOUSLY INJURED BY A
CARELESS FIRING. NEVER FIRE THE BALL WHEN PEOPLE ARE NEAR OR APPROACHING THE
FLIGHT PATH OF THE BALL.
5. Repeat for a total of five trials. Find the average height.
6. Using equations 1, 2, and 3, and your measurements, predict the position of the point on the lab floor
where the ball will land after being fired from the apparatus up on the lab bench table top. To have the
ball miss going into the catcher so it can fly to the floor, the pendulum/catcher must be swung upward
out of the way and held stationary by the ratchet mechanism on the apparatus housing.
7. Tape a piece of magic paper on the floor and mark an “X” where you predict the ball to land. Indicate the
uncertainty in your prediction by marking a most probable range on the paper, like an error bar.
Conclusion: Did your ball land where you predicted? If your ball lands within the MPR of your prediction, you may win
a prize from your teacher or perhaps it will count for a quiz score.
23
8. The Slingshot
Equipment:
two metal rods
meter stick
long rubber band (doubled over when used)
wood block (as used in the friction lab)
the same surface paper used in the friction lab
blue and brown spring scales
Purpose: To derive an experimental value for the effective spring constant of a rubber band, k eff, and compare that with a
value for k determined statically.
W e will launch a wood block from a slingshot made of two rods and a rubber band. Friction, F f , is present
(measured by the spring scale) and after the block is released, it slows to a stop. The maximum displacement of the rubber
band and wood block from their equilibrium position, )x, can be measured (then they would be at rest and ready to be
released). As well, after the block is released and then pushed away by the stretched rubber band, the distance traveled by
the block from its equilibrium position to the block’s final resting position, d, can also be measured. Throughout the lab,
the block is always meant to side only on your lab bench.
Background: Starting from the work-energy theorem, show that if the block stretches the rubber band from equilibrium
by a distance )x and then travels past the equilibrium position a distance d before being brought to rest by
friction, then the effective stiffness of the rubber band, k eff , is given by:
Procedure:
A. The dynamic part:
1. Insert the rods into two adjacent holes on your lab bench. Stretch the doubled rubber band around them (this
means you have four layers of rubber band altogether).
2. Tape the paper down on the table in the region the block will slide.
3. Using the method from the previous friction lab, determine the force of friction, F f , on the block when the
block is pulled at constant velocity by the blue fish scale along the paper (no rubber band here).
Remember to keep the fish scale horizontal as you pull and read the side with the units of force. The
value should be around (0.5 - 0.8) N.
4. Now hold the wood block against the stretched rubber band, mark the position of the block, )x, on the paper.
5. Let the wood block go. After the block slides to a rest, mark the final position of the block.
6. Repeat for a total of five trials. Pull the block to it’s starting position against the rubber band, release, and
mark the block’s final position. Compute k eff from each five trials. Compute the average k and the
standard deviation of the mean (which is the uncertainty in the average value).
B. The static part:
1 . Now check your answer by measuring the spring constant of the rubber band statically. To do this, use the
brown fish scale to pull the block against the rubber band and read the value when the block is in the original
starting position from part 4 above. If pulling the block with the brown scale to the original position puts the
scale out of range, then just pull the block a shorter distance until the force can be measured. Use Newton’s 2 nd
Law to show that the reading on the scale F scale equals the force of the rubber band F RB. Note that the force of
the rubber band obeys Hooke’s Law. Determine a static value for the effective spring constant. Call it k static.
Analysis: Compare your two values of k. Are they the same within the uncertainties?
Repeat for another rubber band configuration but this time choose a different initial position and predict where the block
will stop.
24
9. Centripetal Acceleration
Equipment list:
circular motion apparatus
pan or digital balance
full weight set and hangers
string (the string is stored by wrapping it around the "vertical pointer")
stop watch
meter stick
Purpose: In this lab you will calculate a centripetal acceleration and confirm the value of the force necessary to cause this
acceleration.
Theory: Any body undergoing circular motion must be accelerating even if it is moving at constant speed since the
direction of its velocity vector is continuously changing. Since acceleration is a vector it has a magnitude and a direction.
It can be shown 4 that the magnitude of the centripetal acceleration is v 2/r, where r is the radius of the circle the body is
moving in, and v is the body's speed. The direction of the acceleration vector at any instant is always toward the center of
the circle.
By Newton's 2nd Law, any body that is accelerating necessarily has a non-zero net force acting on it. In circular
motion, the net force is often called the centripetal force, but it is important to understand that the centripetal force is the
name of the net force acting on a body and always represents the vector sum of "real" forces acting on the body. The
centripetal force is not itself a single, "real" force but the name given to the net force acting on a body causing it to move
in a circle.
Leave at least one page in your theory section for further derivations.
Introduction: This lab is performed in two parts. In the first part a "bob" undergoes circular motion when spun by hand
(see the diagram on the next page). By knowing the radius of the bob's circular path and the time it takes to complete one
revolution, you can compute the magnitude of the centripetal acceleration of the bob, a c.
In the second part of the lab, in a non-accelerating static situation, you will measure a force equivalent in
magnitude (but not conceptually equivalent!) to the net force required to make the bob accelerate in the first part. This
force is the weight of a hanging mass, W hanging.
The analysis consists of comparing W hanging to M bob @ a c. These two values should be equal.
Note: The following paragraph is an in-depth explanation of why W hanging is equal to M bob @ a c.
To see that the magnitude of the static force measured in the second part of the lab is equal to the mass of the bob
multiplied by the resultant acceleration of the bob, you must understand that the spring is stretched by the same amount in
both the dynamic and static parts, and you must understand that if a spring is stretched by the same amount in each case,
then the same magnitude of force is acting on it. However in the dynamic part of the lab (part 1), the spring exerts an
inward force on the bob to keep the bob accelerating and this is the only horizontal force acting on the bob. By Newton's
3rd Law then, the bob exerts a force back on the spring outward thereby stretching the spring. In the second part of the lab
(part 2), there are no accelerations, the bob is in static equilibrium. The hanging weight pulls out on the bob via the
connecting string (note: the tension in the connecting string is only equal to the hanging weight since there is no
acceleration) and the spring pulls inward on the bob. Since the acceleration of the bob is zero, the net force acting on the
bob is zero, so the force of the hanging weight on the bob is equal in magnitude to the force of the spring on the bob.
Procedure:
Part 1. The dynamic part.
4
See Giancoli's Physics, 5th edition, pages 113.
25
Preparation: Setting the radius of the circle. W ith the spring not attached to the bob, allow the bob to freely hang without
moving. Position the vertical pointer underneath the bob so the rod and bob match exactly. You now have set the radius of
the circle. Measure the radius and record its value. You might have to move the horizontal position of the counter-weight
and bob assembly in addition to the vertical pointer to achieve the correct alignment. Re-attach the spring to the bob. Use
the leveling screws on the base of the apparatus to level the platform. You are now ready to "run" the dynamic part of the
experiment.
Running part 1. Practice rotating the bob (while the spring is attached to it) by turning the vertical shaft. Make absolutely
certain that the bob is swinging directly over the pointer at all times. If the bob is not directly over the pointer as it swings
in its circle, then the string holding up the bob will have a horizontal component. The horizontal component of the string's
tension force will alter the spring's force on the bob and your results will be inaccurate. As shown in the above diagram,
position your hand near the bottom of the shaft where it is roughened for a good grip. You must rotate the shaft and
therefore the bob at a constant speed. You can check to see if you are doing this because if you turn the shaft at the
"correct" speed then the bob will be just over the vertical pointer. You may have to hold on to the bottom section of the
whole apparatus to keep it from turning with the bob.
W hen you have gained confidence in turning the shaft at constant speed, have your lab partner use the countertimer to time at least fifty total rotations. Call this time the total time, T total. You now have the necessary data to calculate
the centripetal acceleration of the bob.
Record all your measured
values in the data section of
your lab book.
In the theory section
of your lab book (you do have
the space, right?), derive an
equation for the centripetal
acceleration, a c, in terms of
the radius of the circle you
measured and the total time of
all fifty turns. Your final
equation should be a
function of r and T total only,
no speed v! Hint: the
circumference of a circle is
2 Br and the speed of a body
moving in a circle is 2 Br/t,
where t is time for one
revolution. Use your final
equation to find the numerical
value of a c in your calculation
section.
Part 2. The static part.
Leave the spring attached to
the bob. As shown in the
diagram at the right, place an
amount of mass on the hanger
that stretches the spring so
that the bob moves over the
Your setup for the static part.
vertical pointer.
The value of the hanging
mass's weight (include the hanger!) is equal in magnitude to the force that was required to make the bob accelerate in part
1. This equivalence is actually far from obvious but is worth some effort on the student's part to fully explain it.
26
Part 3. The last thing to do is to measure the mass of the bob. From Newton's 2nd Law, to equate the net force and the
acceleration of a body, the mass of the body must be known. A question to answer is whether you should measure the
mass of the spring as well as the bob.
You should repeat the above procedure for one or two more radii. Switch jobs with your partner. Consult your
instructor for more details.
Conclusion: Compare your two values, W hanging and M bob @ a c, and discuss the results.
Questions to consider.
Speculate on possible sources of systematic errors that could be eliminated in future labs.
In terms of systematic errors, would you expect W hanging or M bob@a c to be consistently larger? If you can think of a reason,
explain why.
W hat is the experimental value of using fifty total rotations rather than just one or ten? W hat type of error does this
technique minimize?
In the dynamic part of the lab, if the string tension force on the bob has a horizontal component pulling out on the bob,
will this make the spring force on the bob larger than it should be or smaller than it should be? How would this change if
the string tension force had a horizontal component pulling in on the bob? (Remember, there should be no horizontal
component of the string tension force.)
M ore things to do:
For one configuration (i.e., one pointer position) the hanging mass weight, the radius, and the period are all related by one
equation. W ith this equation in mind, take five data sets of those three parameters for five total configurations and
construct a graph with the appropriate axis that gives a straight line whose slope can be related to the mass of the bob.
Find the slope and calculate and compare the mass of the bob to its measured value found earlier.
27
10. Mass on a Spring
Equipment List:
conical brass springs
support rod
hanger with hook
full weight set and hanger
stopwatch
meter stick
pan or digital balance
Purpose: To determine the spring constant, k, of a given spring using
two different methods.
Theory: A mass on a spring is a good example of a simple harmonic
oscillator. Simple, because it has one frequency, harmonic, because the
equations of motion can be solved with sines and cosines, and an
oscillator because the motion is repeated. W hen a force is proportional
to the negative of the displacement it is called a restoring force; the force
always accelerates the body back to its equilibrium position. This is the
case with a mass on a spring. Stretch the mass, and the spring seeks to
pull the mass back to equilibrium. Compress the mass against the spring,
and the spring seeks to push the mass back to equilibrium. An object
under the influence of a restoring force exhibits simple harmonic motion
(SHM). From a physics point of view, a mass can be suspended
vertically by a spring in two different ways: that is, a mass can be
hanging from a spring and not accelerating (it is in static equilibrium) or
continuously bobbing up and down always accelerating (dynamically
speaking, it is oscillating).
Recall Hooke’s Law:
F = - kx
In the static case,
F net = 0
F g - F s = 0 (W here here the minus sign is taken care of
by the direction of the vectors)
mg - kx = 0
thus,
mg = kx
In the dynamic case,
F net = ma
Even if gravity acting on the system, it is a constant force and can be replaced with C. Then,
1)
-kx = ma + C
This equation has the solution:
2)
x = A sin Tt
W here A is the amplitude or maximum displacement
T is the angular frequency
x is the displacement at any time
t is the time
The time for one complete oscillation is called the period, T. And the linear frequency, f = 1/T.
28
The angular frequency is T = 2 Bf
By combining equations 1 and 2, one finds that the angular frequency can be written in terms of the spring
constant, k, and the mass, m. That is, T2 = k/m.
Procedure: Hang the spring, large side down from an arm attached to the support rod. (Note: There may not be sufficient
hooks to go around, in which case, you will have to improvise.)
Static:
Tape the meter stick, low numbers on the top, next to the rod.
Measure the mass of the hanger. Record.
Determine the equilibrium position of the spring. (You may use whatever is easiest for you. For example, the
bottom of the spring or the bottom of the hanger.) Record.
Add some mass to the hanger. Perhaps 5 - 20 g. Better data will have a larger range, to a point! The spring can
get deformed and then your value for k will be non-linear. Don’t add too much weight! Record the mass
and the new equilibrium position. Write down the number you read directly from the meter stick.
Repeat for a total of five data points. This will be the data for the static part of the experiment.
Dynamic:
You may choose different masses for this part of the experiment if the masses from the static part are too light or
too heavy. Play around to see what five masses will give you sufficient spread of the data without
damaging the spring.
Add mass to the hanger. Set the mass oscillating. Time 10 oscillations. Record the total mass (added + hanger)
and the time t 10
Repeat for a total of five data points. This will be your data for the dynamic part of the experiment.
Data: You should have five pairs of data points for the static part and five pairs of data points for the dynamic part.
Divide t10 by 10 to find the period of one oscillation, T.
Analysis: Construct two full page, linear graphs, one for the static experiment and one for the dynamic. The slope of each
should be related to k. For the dynamic graph, combine T2 = k/m. T = 2 Bf and f = 1/T to find a relation between your
experimental data for m and T and the spring constant, k. Remember! Your graph must be linear to be valid!
Find the slope of each graph. Find k, the spring constant, from your two values, static and dynamic, for the slope.
Do a discrepancy test. W hich of your experimental values will you consider “known” and why?
Conclusion: State your two values for k and the discrepancy. Do your values agree? W hy or why not?
29
11. The Pendulum
Equipment List:
two meter length of string
support rod with clamps
full weight set with hangers
stopwatch
meter stick
Purpose: By a graphical method, determine the value of g (the gravity field or in free fall, the acceleration of a body) by
use of a simple pendulum, and then use the graph to extrapolate the most practical length for a “pendulum clock”.
Theory: The swinging of a pendulum, repeating the same motion at regular intervals of time (a time called the
period, T). for small enough amplitudes, the period of a simple pendulum depends on the length of the pendulum and the
gravity field it is in, but not on the mass of the bob. In a real pendulum there is a slight dependence of the period on the
amplitude or initial angular displacement but for angles of less than ten degrees this is negligible.
Procedure: Connect the hanger to the support rod such that the pendulum will be able to swing freely all the way to the
floor. Choose a length of string and measure it. Record this as the initial length of pendulum Attach a mass on the end.
Convince yourself that the period of the pendulum is independent of the mass. Set the pendulum in motion. Time five
cycles. Record this as t5. Repeat for a total of five different lengths of string. Remember a large range of values will
decrease the uncertainty of your result.
Analysis: Construct a linear, full-page graph using your data whose slope is g. First, divide t5 by 5 to obtain the period,
T. Determine T2 from your values of T. Plot pairs of T2 and l. Determine the slope of the graph and the uncertainty of
the slope. Determine the value of g, and your uncertainty from the slope of the graph.
Using your graph, can you estimate what length would be best for a pendulum clock. (Note: you might want to
decide what the period would be and then calculate, given that period, what T2 is.) Verify your estimate experimentally.
Conclusion: State your experimental value for g with uncertainty. Did your estimate for the length of the pendulum clock
prove viable. Why or why not?