Download Work and Energy - Student Worksheet

Work and Energy
A. Tomasch 01/10
Pre-Lab Question
If moving objects have energy due to their motion (called Kinetic Energy (KE)),
what is the source of kinetic energy for a ball that starts at rest and rolls down a hill?
EXPLORATION
Exploration Materials
Dedicated Components
1 Pitsco PC Sportster car
1 ½ inch x 2” steel hex bolt with nut
1 10’ steel wall stud ramp
8 ½ inch cut steel washers
1 Homer bucket ramp support
1 Guide Wire Assembly
1 spring scale– Ohaus 2000g/20N
1 clear plastic ruler
1 meter stick
1 digital scale
1 stop watch
1 calculator
1 2” long thin rubber band
Mechanical Work
Pitsco Car Number______________
For the following series of experiments, load your car with the
bolt and eight washers, seven washers on the top and one on
the bottom of the car.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
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Definition: If a force acts on an object parallel to its direction of motion, the
product of the force and the distance the object moves is called
mechanical work. Work is a scalar quantity which can be positive, negative
or zero. Mechanical work is positive if the force acts in the direction of
motion and negative if it acts in opposition to the direction of motion. A
force perpendicular to the direction of motion does zero mechanical work.
A zero net force also does zero work.
Definition: Energy is defined as the ability to do mechanical work.
1. Using your spring scale, determine the force due to gravity on the car acting
down the incline of your ramp. Use a long rubber band to attach the scale
to the small metal loop at the rear of the car. Drag the car slowly up the
ramp at a constant speed and read the scale. Your scale reads in grams
and we want to state forces in Newtons, so you must convert grams to
kilograms and multiply by the acceleration of gravity (g = 9.8 m/s2) get the
force in Newtons. Show your calculation.
Force of Gravity Acting Down the
Ramp (N)
2. Assuming that the length of the ramp is 2.74 meters, use your answer
to question 1 to estimate the amount of work done by gravity on the car as it
rolls down the ramp. A force of one Newton pushing on an object parallel to
its direction of motion for one meter does one Joule of work. Show your
calculation.
Work Done by Gravity for a Car
Rolling Down a Ramp (Joules)
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3. How does the work done by gravity on a car released half way up the ramp
compare to the work done by gravity on a car released at the top of the
ramp? Estimate the amount of work in Joules for a car released half way up
the ramp. Show your calculation.
Work Done by Gravity for a Car
Rolling Half Way Down a Ramp
(Joules)
If we release a car from rest at the top of a ramp we can measure the
final speed of the car at the bottom of the ramp by measuring the
average speed of the car. The average speed (m/s) is length of the ramp
(m) divided by the time it takes the car to roll to the bottom (s):
Average Speed = (Ramp Length)/(Time to Roll Down Ramp).
Since the car starts at rest (zero initial speed):
Average Speed = ½ (Initial Speed + Final Speed)
Average Speed = ½ (0 + Final Speed) = ½(Final Speed).
So the final speed of the car at the bottom of the ramp is simply twice
the average speed:
Final Speed = 2(Average Speed)
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4. Release your car from rest approximately half way up the ramp and measure
the time to reach the end of the ramp with the stop watch. Compute the
average speed of the car as half the ramp length divided by the measured
time. Estimate the final speed of the car at the bottom of the ramp from the
measured average speed. Show your calculation for one trial.
Distance = ½(2.74 m)
Time
(s)
Trial #1
Trial #2
Trial #3
Average Speed
(m/s)
Final Speed
(m/s)
Average Final Speed
5. Repeat the experiment, but this time release the car from the top of the
ramp and again compute the average speed as the total length of the ramp
divided by the time. Again estimate the final speed of the car at the bottom
of the ramp. Show your calculation for one trial.
Distance = (2.74 m)
Time
(s)
Average Speed
(m/s)
Final Speed
(m/s)
Trial #1
Trial #2
Trial #3
Average Final Speed
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6. Qualitatively, how is the average speed down the ramp related to the
distance the car rolls down the ramp?
7. Qualitatively, how is the average speed down the ramp related to the
amount of work done by gravity as the car rolls down the ramp?
Kinetic Energy and Gravitational Potential Energy
Definition: Kinetic Energy (KE) is defined as one half the product of an
object’s mass m and its speed v squared:
KE ≡ ½
mv2
Definition: Gravitational Potential Energy (GPE) is defined as the product
of an objects weight W and its height h above an arbitrary horizontal
reference elevation:
GPE ≡ mgh = Wh
All heights must be measured consistently relative the chosen reference
elevation when calculating GPE.
8. Determine the mass of your car using one of the digital scales provided and
record it here in kilograms (show your conversion from grams to kilograms):
Mass of Car (kg)
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9. Let’s choose the bottom of the ramp to be our reference level for
calculating gravitational potential energy, where we will define the height
zero. Hold the car in its starting position at the top of the ramp. With your
meter stick measure the height of the top surface of the ramp above the
table top where the rail rests on the edge of the support bucket. Next
measure the height of the top surface of the ramp where the ramp enters
the bottom “car catcher” bucket. The difference between these two
measured heights is the distance the car will descend vertically as it rolls
down the ramp, that is, how much higher the car is at the top of the ramp
than at the bottom. Record all your measurements in the table below and
convert the final height difference to meters. Show your final conversion
from centimeters to meters.
Top Height of Ramp at Car Center (cm)
Bottom Height of Ramp at Car Center (cm)
Difference Between Top and Bottom Height
of Ramp (cm)
Difference Between Top and Bottom Height
of Ramp (m)
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10.Calculate the gravitational potential energy of your car’s mass using the
difference in height determined in question (9) and report it along with the
work done by gravity on the car you calculated in question (2). Also calculate
the kinetic energy of your car at the bottom of the ramp using the average
final speed you determined in (5) for the car rolling the entire length of the
ramp. Show your calculations (g = 9.8 m/s2).
Gravitational Potential
Energy at Top of Ramp
(Joules)
Work Done by Gravity
Rolling Down Ramp
(Joules)
Final Kinetic Energy of
Car at Bottom of Ramp
(Joules)
11. How does the number of Joules of work done by gravity that you estimated
in question (2) compare with the gravitational potential energy and the
final kinetic energy you calculated in question (9)?
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12. Can we use gravitational potential energy as an alternative to measuring the
force down the ramp and the length of the ramp to account for work done by
gravity? If we define gravitational potential energy to be zero at the bottom
of the ramp, do your results support the conclusion that total energy
(gravitational potential energy plus kinetic energy) is conserved? Explain.
Theorem:
The Work-Energy Theorem states that the work done on an
object by a net force Fnet acting parallel to a displacement d equals the
change in the object’s kinetic energy:
ΔKE ≡ Fnet x d = Wnet
ΔKE ≡ KEfinal – KEinitial = ½ mvf2 - ½ mvi2
If positive work is done on an object (the net force acts in the direction of
motion), its speed increases.
If negative work is done on an object (the net force acts in opposition to the
direction of motion, its speed decreases.
If zero net work is done on an object its speed remains constant.
Assuming the car starts at rest (zero initial speed) the Work-Energy
Theorem predicts that the final kinetic energy at the bottom of the ramp is
½ (Mass of Car)×(Final Speed)2. At the top of the ramp the gravitational
potential energy is the product of weight and height = (Mass of Car)×(Ramp
Height)×(Acceleration of Gravity).
In symbols we can state these relationships very simply:
2
KE  21 mvfinal
GPE  mgh
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13. Let’s apply the Work-Energy Theorem to our car rolling down the ramp,
assuming that only gravity does work on the car. Using the expressions for
kinetic energy (KE) and gravitational potential energy (GPE) given above,
conserve energy, that is, set the final kinetic energy equal to the initial
gravitational potential energy at the top of the ramp (that is, at height h)
and solve for the final speed of the car. Show your calculation below.
14. Does the result of your calculation in (13) depend on the mass of the car?
Why? Is this consistent with the results we obtained previously for cars of
different masses rolling down the ramp? What single factor determines the
final speed at the bottom of the hill, assuming that only gravity does work on
the car?
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15. Using the difference in height you determined for your car at the top of the
ramp in (8) and the expression you derived in (12) calculate the speed of
your car at the bottom of the ramp and compare it to the average final speed
you measured at the bottom of the ramp in (5) for the car rolling down the
entire length of the ramp. Show your calculation below (g=9.8 m/s2).
Calculated Final Speed at the
Bottom of the Ramp (m/s)
Measured Final Speed at the
Bottom of the Ramp (m/s)
16. Our work-energy analysis of the car’s motion assumes that only the force of
gravity does positive work on the car and all of the energy gained by the car
contributes to increasing its final speed. Cite one additional force that acts
on the car that does negative work on the car and therefore causes the car
to be moving more slowly at the bottom of the ramp.
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17. Cite one additional way the car has gained kinetic energy that we have not
accounted for in our analysis. Will this make the car move faster or slower at
the bottom of the ramp? (Hint: What parts of the car are moving and
therefore have kinetic energy we have not accounted for?).
Application
Materials
Dedicated Components
1 Hot Wheels Car
1 calculator
Shared Components
1 8’ steel wall stud ramp with Hot Wheels Track
2 Homer bucket ramp supports
1 Homer bucket car-catcher
1 50 gram digital scale w/pulley and Homer Bucket Support to measure
force of gravity down the ramp
1 clear plastic ruler
1 meter stick
1 digital scale for weighing cars
Packing tape to secure ramp
Data acquisition system for cars: ring stand photo gate support, laptop PC running Logger
Pro “speed trap” application and MS Excel
Conservation of Energy
We will now make a quantitative study of energy conservation for Hot Wheels
cars rolling down a ramp. We will collect our data together as a class using the
same “speed trap” that we used to study Newton’s Second Law with the rocketpowered Pitsco cars to measure the speed of cars at the bottom of the ramp.
We’ll enter our measurements directly into an Excel spreadsheet and discuss the
results as a class. The spreadsheet results will be posted on CTools for you to
print out and include in your lab notebook. Record the data for your car below.
Be sure to convert the car mass in grams to kilograms and the force parallel to
the ramp to Newtons. Show your conversions for grams to kilograms and grams
to Newtons of force below. Finally, please report your values to the
entire class on the front black board.
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Data: g=9.8 m/s2
Hot Wheels Car Number________
Car Mass_________ (g) Car Mass_________ (kg)
Force of gravity Parallel to Ramp Framp ___________ (g)
Force of Gravity Parallel to Ramp Framp ___________ (N)
Time Between Photo Gates Δt __________________(s)
Speed at Bottom of Ramp = (0.35 m)/ Δt =_____________m/s
18.The Hot Wheels cars will descend a height of 0.56 m as they roll down the ramp.
Use the expression you derived in (13) to predict the final speed of the Hot Wheels
cars, assuming that only gravity does work as they roll down the ramp. Show your
calculation (g= 9.8 m/s).
Predicted Final Speed at Bottom of Ramp (m/s)
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Record the data for the class measurements below:
Car Number
Car Mass
(kg)
Force of
Gravity
Parallel to
Ramp
(g)
Force of
Gravity
Parallel to
Ramp
(N)
Time
Between
Photo
Gates Δt
(s)
Measured
Final Speed =
(0.35 m)/ Δt
(m/s)
#1
#2
#3
#4
#5
#6
#7
#8
Average Final Speed at the Bottom of the Ramp (m/s)
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19.Compare the average measured final speed for the class measurements to the final
speed you predicted in (18). In light of your answers to (15) and (16), discuss
what might account for any difference you observe. You may also suggest
additional factors that might cause the predicted and measured values to differ,
but be specific.
20.Based on your measurements and analysis, can work-energy analysis be used as
an alternative to Newton’s Second Law (F = ma) to predict the final speed of a car
rolling down a ramp? Give your overall assessment of how well your measured
values and calculations support this conclusion.
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Everyday Applications
APPLICATION
1. Airplane pilots have an expression “Altitude is speed.” Explain what this means
using the concepts of gravitational potential energy, kinetic energy and energy
conservation. Neglect any energy lost to air resistance.
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Summary:
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A net force which acts parallel to an object’s direction of motion does
mechanical work on the object.
The work done by individual forces can be added together to get the work done
by the net force.
Positive work is done when the force acts in the direction of motion.
Negative work is done when the force acts in opposition to the direction of
motion.
Zero work is done if a force acts perpendicular to the direction of motion or if
the net force is zero.
Kinetic Energy (KE) is energy due to motion and is equal to half the product of
an object’s mass and multiplied by its speed squared: KE ≡ ½ mv2
A conservative force does the same work between any two points in space
irrespective of the path taken between the points, or equivalently, zero work
for a closed loop path in space. For conservative forces it is possible to
characterize the work done by the force by means of a potential energy
function.
Gravitational Potential Energy is defined as the product of an object’s weight
and its height above an arbitrarily chosen reference height: GPE ≡ mgh = Wh.
The work done by gravity can be accounted for by differences in the
gravitational potential energy, and hence changes in an object’s height. This
is possible because gravity is a conservative force.
The Work-Energy Theorem states that the work done by the net force is equal
to the change in an object’s kinetic energy.
If positive work is done on an object, its speed increases.
If negative work is done on an object, its speed decreases.
If zero work is done on an object, its speed remains constant.
If only gravity does work the final speed of an object descending a height h down
a frictionless ramp is given by
vfinal = 2 gh
Cleanup
Please attach the Washers and bolt to your Pitsco car, one washer on the bottom and
seven on top.
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Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan 48109
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