PHY132
Experiment
6
Work and Energy
In this experiment you will investigate the work done under a constant force as well as a variable force.
Recall from the lecture that in the absence of friction, work can be calculated using:
W  Fav  x
where F is the constant force (if it is constant, then its average value is equal to its instantaneous value) and
x the displacement of the object. You will measure the constant force and calculate the work done.
If the force is not constant, we can still calculate work using the integral:
𝑥2
𝑊 = ∫ 𝐹𝑥 𝑑𝑥
𝑥1
This is the area under the force vs. position graph. You will use a spring, which is an example of a material
with variable force, and calculate the work done by solving the integral mathematically, and by using the
software to integrate the area.
PART 1
Work done by a constant force
1. You will measure the work needed to lift an object vertically upward at constant speed. The work done
can be calculated using the displacement and the average force.
2. Connect the Motion Detector to DIG/SONIC 2 of the LabPro. Connect the Vernier Force Sensor to Channel
1 of the interface. If your sensor has a range switch set it to 10 N.
3. Go to folder from Probes and Sensors. Then open the folder Force Sensors and next the folder DualRange Force Sensor. Now select the file Motion and DFS 10N. Three graphs will appear on the screen:
position vs. time, velocity vs. time, and force vs. time.
4. You want to change a few axes first. Click on top of the word Velocity (y-axis) in the second panel.
Select “Force” instead of “Velocity” for the y-axis. Click on the word Time (x-axis) of the second panel.
Select “Position” instead of “Time” for the x-axis. Now you have a graph force vs. position.
5. It is necessary to calibrate the force sensor.
6. Choose Calibrate from the Experiment menu. Select CH 1 Force Sensor. Select the Calibrate tab. Click
.
7. Remove all force from the Force Sensor. Enter a 0 (zero) in the Reading 1 field. Hold the sensor
vertically with the hook downward, make sure it is stable, do not move. Click “Keep”. This defines the
zero force condition.
8. Hang the 500-g mass from the Force Sensor. This applies a force of 4.9 N. Enter 4.9 in the Reading 2
field, and after it is stable, click “Keep”. Click
to close the calibration dialog.
9. Check your calibration. Hang some other weight from the force sensor and check whether the reading in
newtons is correct (reading will show on the top left-hand corner).
1
10. Hang a 200-g mass from the Force Sensor.
11. Place the Motion Detector on the floor or on the table, but away from
table legs and other obstacles.
12. Hold the Force Sensor and mass about 0.2 m above the Motion
Detector. Click “Collect” to begin data collection. Wait about 1.0 s
after the clicking sound starts, and then slowly raise (with constant
speed) the Force Sensor and mass about 0.5 m straight upward. Then
hold the sensor and mass still until the data collection stops at 10 s.
Dual-Range
Force Sensor
13. Make your own Date Table 1. Remember to include uncertainties
when recording your measurements (or prepare a table of
uncertainties).
14. Examine the position vs. time and force vs. time graphs by clicking
the Examine button. Identify when the weight started to move
upward at a constant speed and when the weight stopped moving
upward. Select this region.
15. Important: Why do you select the region of constant speed? Draw a
free-body diagram of the object exactly as you do when solving
problems. What is the force with which you lift the mass when you lift
at a constant speed?
16. Record the initial time, final time (for reference) and initial height
and final height in your data table.
17. According to the region selected, determine the average force exerted while you were lifting the mass.
You have learned how to use the software analysis tools by now. Record the value in your data table.
18. You should not include the brief periods when the up motion was starting and stopping. Why not?
19. Change the mass and repeat for a total of 3 different masses.
20. Save only one graph and include a printout with your report: make sure to show any toolbox you have
used.
ANALYSIS OF PART 1
1.
Make your own Analysis Table 1 the way you have learned previously.
2. Calculate the displacement of the mass.
3. Use the average force you obtained and calculate the work done by using Waverage = Fav  y.
4. You can also get Work using the software. Click on the force vs. position graph. Corresponding to the
interval you have chosen (carefully verify the initial and final positions are the same as in the
previously selected region), make that same selection in the force vs. position graph, and click the
Integrate button, . The value you get (Warea) should match the work measured before.
5. Compare Warea or Waverage with an expected value. The expected work (Wexpected) is calculated using the
same y as measured before, but using F as F = mg.
6. Remember to calculate the percent difference for all three masses; present it in an organized way in the
analysis table. Always include errors.
2
PART 2
Work done by a variable force
Unlike the work needed to lift a mass, the work done in stretching a spring is not constant. The work can still
be calculated using the area under the force vs. position graph.
1. Recalibrate the force sensor, as it will be used in a new position. You must always check the calibration
during experiments.
2. Attach one end of the spring to a rigid support. Attach the Force Sensor hook to the other end. Rest the
Force Sensor on the table with the spring extended but relaxed, so that the spring applies no force to the
Force Sensor.
3. Place the Motion Detector about one meter from the Force Sensor, along the line of the spring. Be sure
there are no nearby objects to interfere with the distance measurement (clean your work space!).
Force Sensor
Force Sensor
Motion
Motion Detector
Detector
Dual-Range
Force Sensor
4. Using tape, mark the position of the leading edge of the Force Sensor on the table. The starting point is
when the spring is in a relaxed state (position of equilibrium). Make the initial position be the position
of equilibrium of the spring. Hold the end of the Force Sensor that is nearest to the Motion Detector as
shown above. The Motion Detector will measure the distance to your hand (that is fine), not the Force
Sensor. With the rest of your arm out of the way of the Motion Detector beam, click
.
5. Click Collect. Within the limits of the spring, slowly (and with constant speed) move the Force Sensor
and stretch the spring about 30 cm or more, over several seconds. Hold the sensor still until data
collection stops. Do not get any closer than 20 cm from the Motion Detector.
6. Make your own Data Table 2 for this part.
7. Identify the time when you started to pull on the spring and the time you stopped. Record these as
initial and final times in your data table, for reference. Record also the initial and final positions.
8. The difference between the initial and final positions is the amount of stretch in the spring. Record this
as your displacement x.
9. Click on the force vs. position graph. Corresponding to the time interval you have chosen, select that
part of the graph. Then click the Linear Fit button to determine the slope of the force vs. position
graph. The slope is the spring constant, k. Write a few equations to verify that. Record the slope and
intercept in your data table.
Spring forces are governed by Hooke’s law, which states 𝐹⃗𝑠 = −𝑘𝑥⃗. This equation represents a
straight line with a negative slope. If your plot F vs. x shows a positive slope, explain why.
10. Save only one graph and include a printout with your report: show any toolbox you might have used.
3
ANALYSIS OF PART 2
1. The area under the force vs. position graph is the work done to stretch the spring. Select the region
over which you stretched the spring, and click the Integrate button to determine the area. Record this
as work done (Warea) in your Analysis Table 2.
2. You can also calculate the work done by a spring using the formula Wequation = ½ k(𝒙𝟐𝟐 − 𝒙𝟐𝟏 ), where
x is the deformation in respect to the position of equilibrium (this is the result of the integral
presented on the first page). Calculate the work done this way and record it in your table.
3. Remember that you must keep track of all uncertainties, and propagate the errors. You can perform
the final error propagation at home, as long as you leave class knowing all formulas and values
necessary.
REPORT DUE IN 1 WEEK
For next week you must submit a lab REPORT with propagated errors displayed in the Analysis Table
(and detailed calculations shown in the Analysis Section). Recall your previous labs and apply whatever
is applicable.
Attach the following Video Analysis exercise sheet to the end of your lab report.
4
VIDEO ANALYSIS
The Net Work–Kinetic Energy Theorem can be derived from Newton’s 2nd Law. If the velocity of an
object is measured, this theorem can be used to calculate the net work on an object when the force on it is
not constant. The equation for the theorem is:
𝒙𝟐
𝟏
𝑾𝒏𝒆𝒕 = ∫ 𝑭𝒏𝒆𝒕𝒙 𝒅𝒙 = 𝟏𝟐 𝒎𝒗𝟐𝒙𝟐 − 𝟐𝒎𝒗𝟐𝒙𝟏 = 𝚫𝑲
𝒙𝟏
Notice that this theorem relates 𝚫𝐊 with the work done by the net force, and not just any force. We will
check if the Net Work–Kinetic Energy Theorem holds from one arbitrary position to another, even if
friction is present.
We will choose x1 = 0.199 m and x2 = 0.607 m.
1. Open the Logger Pro experiment file <NetW-KE.cmbl>. This file opens the movie <NetW_KE.mov>.
The movie has been synchronized with data recorded by the motion detector and the force sensor.
Since the mass of the cart and its affixed force sensor is known to be 1.216 kg, the Logger Pro new
column feature has already been set to calculate vx (the x-component of the cart’s velocity) and K (the
kinetic energy of the cart) in each frame.
2. Watch the events in slow motion by clicking on the Start button in the Replay box.
3. If friction is present and the cart is being pulled, there are four forces on the cart. List these forces:
Vertical Forces: ______________________
Horizontal Forces: ______________________
______________________
______________________
4. What would 𝑭𝒏𝒆𝒕𝒙 be? _____________________________________________________________
5. Near the left end of the top Toolbar, switch from Page 1: “Data” to Page 2: “Analysis”. Examine the
graphs: you will use them to answer the next questions.
6. Find the value of the cart’s kinetic energy change in Joules for the chosen range of motion. Perform
your calculations using three decimal places. Show equations and calculations, not just final answers.
5
7. Next, using the definition of work and software tools, find the value of the work on the cart in Joules
between positions x2 and x1. Show equations and calculations.
8. Compare your answers: calculate the percent difference between answers to questions 6 and 7. Does
the Net Work–Kinetic Energy hold reasonably well?
6