Physics 161
Work and Energy
Introduction
Work is defined as the product of the force used to move an object times the distance the object
is moved. In this lab students will get familiar with calculating work done by constant, linear and
varying forces. In all parts, students will use a motion sensor to measure the distance that a cart is
traveling and a force sensor to determine the force that pulls the cart on the track. Students will
also plot the force vs distance in the Data Studio to determine the amount of work by finding the
area under each curve. In part I, the cart will be pulled by a constant force created by a dropping
mass. In Part II, the mass will be replaced by a stretched spring. The spring will exert a linear
force on the cart. In part III, the string will be replaced by a falling chain. Here, the force that
pulls the cart changes while the chain’s mass linearly decreases as the chain makes contact with
the ground. For the last part, part IV, the chain will fall in a container filled with water. As the
chain falls into the water, the resistive force of water will have a non-linear effect on the force
that pulls the cart.
Reference
Young and Freedman, University Physics, 12th Edition: Chapter 6, section 6.1-6.3
Theory
In your physics text, the work done by a constant force is defined as “the product of the
magnitude of the displacement times the component of the force parallel to the displacement.”
W  F|| d
(1)
W  Fd cos( )
(2)
Or
If the force is constant, a graph of F vs. time should be a horizontal line. Gravity is a
conservative force, so if it is the only constant force working on the cart, then we expect the total
work to be constant. (For this lab, we will assume that the force of friction is negligible.) Ideally
the motion would be measured at the center of mass. However, for all parts the measurement
taken by the motion sensor will be from the front of the cart.
On the other hand, if the force acting on the object is not constant then “the work done by a nonconstant force in moving an object between two points is equal to the area under the F|| (F
cos(θ)) versus distance (l) curve between those two points” as shown in Figure 1.
b
b
W  F|| dl  F cos( )dl
a
a
(3)
Figure 1: Fcos(θ) vs distance plot
Our objective in this lab is to calculate the work done by forces:
a) Using the numerical integration method
b) using an appropriate trend line to find an analytical expression for force and then
calculate a definite integral using this force.
Calculation of the area under the curve: The Trapezoid rule (or Trapezium rule) is one way of
calculating definite integral. In many cases where an analytical function is not available, and
only observed data is at hand, this rule can be applied to the data in order to calculate the area
underneath the curve. The area underneath the curve can be segmented into trapezoids with equal
widths as shown in Figure 2. The total area under the curve can be approximated as the sum of
the areas of the trapezoids.
Figure 2. Trapezoid rule
The area of the first shaded trapezoid = A1=Δx1(f(a) +f(x1))/2. Similarly, the area for the second
shaded trapezoid = A2 = Δx2(f(x1) +f(x2))/2. The last area= A3= Δx3(f(x2) +f(b))/2.
The total area is:
A  A1  A2  A3 ....  Ai
i
(4)
Procedure
Part I: Work done by a constant force.
1. To collect data using the setup shown in Figure 3, set up the Data Studio interface with a
motion sensor set to take data at 50 Hz and the switch at the top of the motion sensor set
to record motion at short distances. Hold the track firmly so it will not recoil when the
cart is launched.
Force Sensor
Motion Detector
Dynamics Cart
Track
Figure 3. Setup for Part I experiment
2. Connect the force sensor to the cart and set the trigger rate of the force sensor to 50 Hz as
well. Place the cart on the track and have your lab partner hold the cart on the track
firmly.
3. Connect a piece of string to the force sensor and run the other end of the string over the
pulley (which is attached to the other end of the track). Now place a 100g mass to the end
of the string and leave it hanging over the pulley.
4. Now switch position with lab partner and have him/her stand at the end of the track
where the pulley is. Place the cart 15 cm away from the motion sensor.
5. Press the Start button and then push the tare button on the force sensor. Release the cart.
The cart will be pulled by the mass hanging and will get to the end of the track. Before
the cart hits the end of the track, have your lab partner stop it. You may need to do this a
couple of times to get good data.
6. Press the Stop button after the cart has reached the end of the track.
7. Make graphs in Data Studio of the Force vs. position. To do so, first make a graph of
Force vs. time and then drag the position data to the plot area next to the axis. As you get
close to the axis you should notice a dashed box around the x-axis. As soon as you
release the mouse button, the x-axis should read position instead of time.
8. You want to have a force vs. position graph that looks somewhat like Figure 4 (but the
force values should be negative). Use the Sum button to find the average force between
the distances of 20 to 80 cm.
Figure 4. Work done by constant Force
9. Calculate the work using equation 2 and the value of the force found in step 8. Assume
cos(θ)=1. Call this work theoretical work.
10. Transfer the data for Force vs position to Excel and calculate the work using equation 4.
Call this work experimental work.
11. Calculate the %diff between theoretical and experimental work.
Part II: Work done by a linear force (spring).
12. Replace the mass by a spring. Stretch the spring and connect the other side of it to a 1kg
weight on the floor as shown in Figure 6. Make sure that the stretched spring is
completely vertical.
Figure 6. Work done by a spring apparatus.
13. Repeat steps 4-7 of above procedure. Your plot should look like the plot in Figure 7.
Figure 7. Force vs. Distance for a spring.
14. Use the linear fit to find a line that fits the data the best from 20 to 80 cm position.
15. Calculate the work using equation 3 and the line found from step 14. Call this work
theoretical work. (Remember the limit of the integral is from .2 to .8.)
16. Transfer the data for Force vs. position to Excel and calculate the work using equation 4.
Call this work experimental work
17. Calculate the %diff between theoretical and experimental work.
Part III: Work done by a linear force (chain).
18. Now replace the spring by a chain as shown in Figure 8.
Figure 8. Apparatus set up for the linear force using a chain
19. Repeat steps 4-7 of above procedure. Your plot should look like the plot in Figure 7.
20. Use the linear fit to find a line that fits the data the best from 30 to 90 cm position.
21. Calculate the work using equation 3 and the line found from step 20. Call this work
theoretical work. (Remember the limit of the integral is from .3 to .9.)
22. Transfer the data for Force vs. position to Excel and calculate the work using equation 4.
Call this work experimental work
23. Calculate the %diff between theoretical and experimental work.
Part IV: Work done by a varying force.
24. Place a container filled with water underneath the chain. Repeat steps 4-7 of above
procedure. Your plot should look like the plot in Figure 9.
Figure 9. Plot of F vs. position for a chain falling in the water.
25. Use the Polynomial fit to find a polynomial that fits the data the best from 30 to 90 cm
position. The equation of the polynomial is of the form
P ( x )  A Bx Cx 2 Dx 3
(5)
26. Calculate the work using equation 3 and the polynomial found from step 25. Call this
work theoretical work. (Remember the limit of integral is from .3 to .9.)
27. Transfer the data for Force vs. position to Excel and calculate the work using equation 4.
Call this work experimental work
28. Calculate the %diff between theoretical and experimental work.
Physics 161
Work and Energy Question Sheet
Name:
Partner:
Section #
Please answer all the questions on this sheet. Make sure you perform all the integrations. Include
a sample calculation for finding the area of the first trapezoid from the table.
Part I
1. What was the average force?
2. Calculate the work using equation 2 and the above value.
3. What was the value of the work calculated from Excel?
4. Explain how the graph of Force vs. position looked like? (Explain the physics only.)
5. Did you expect the force to be constant? Why?
6. Calculate the %difference between the theoretical and experimental work.
Part II
7. What was the equation for the linear force?
8. Calculate the work using equation 3 and the above equation.
9. What was the value of the work calculated from Excel?
10. Explain how the graph of Force vs. position looked like? (Explain the physics only.)
11. What was the purpose of using a spring in this experiment?
12. Calculate the %difference between the theoretical and experimental work.
Part III
13. What was the equation for the linear force?
14. Calculate the work using equation 3 and the above equation.
15. What was the value of the work calculated from Excel?
16. Explain how the graph of Force vs. position looked like? (Explain the physics only.)
17. What was the purpose of using a chain in this experiment?
18. Calculate the %difference between the theoretical and experimental work.
Part IV
19. What was the equation for the polynomial force?
20. Calculate the work using equation 3 and the above equation. Show the integration.
21. What was the value of the work calculated from Excel?
22. Explain how the graph of Force vs. position looked like? (Explain the physics only.)
23. What was the purpose of the container with water?
24. Calculate the %difference between the theoretical and experimental work.