Homework answers / question archive / Lab 1: Torsion Pendulum PHYS 2306 Objective To demonstrate that the motion of the torsion pendulum satisfies the simple harmonic form in Equation (3) To show that the period (or angular frequency) of the simple harmonic motion of the torsion pendulum is independent of the amplitude of the motion To make measurements to demonstrate the validity of Equation (6), which relates the angular frequency of motion to the torsion constant and the moment of inertia of the torsion pendulum Recommended background reading x    Knight, Chapters 12, 15  x    CAPSTONE APPENDIX: www

Lab 1: Torsion Pendulum PHYS 2306 Objective To demonstrate that the motion of the torsion pendulum satisfies the simple harmonic form in Equation (3) To show that the period (or angular frequency) of the simple harmonic motion of the torsion pendulum is independent of the amplitude of the motion To make measurements to demonstrate the validity of Equation (6), which relates the angular frequency of motion to the torsion constant and the moment of inertia of the torsion pendulum Recommended background reading x    Knight, Chapters 12, 15  x    CAPSTONE APPENDIX: www

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Lab 1: Torsion Pendulum

PHYS 2306

Objective

To demonstrate that the motion of the torsion pendulum satisfies the simple harmonic form in Equation (3)

To show that the period (or angular frequency) of the simple harmonic motion of the torsion pendulum is independent of the amplitude of the motion

To make measurements to demonstrate the validity of Equation (6), which relates the angular frequency of motion to the torsion constant and the moment of inertia of the torsion pendulum

Recommended background reading

x    Knight, Chapters 12, 15 

x    CAPSTONE APPENDIX: www.phys.vt.edu/~labs/CAPSTONE (this website is case sensitive).

Introduction    Figure 1. 1

A diagram of a torsion pendulum is shown in Figure 1. 1.  A disk with a moment of inertia I₀ is fastened near the center of a long straight wire stretched between two fixed mounts.  If the disk is rotated through an angle  and released, the twist in the wire rotates the disk back toward equilibrium.  It overshoots and oscillates back and forth like a pendulum, hence the name.  Torsion pendulums are used for the timing element in some clocks.  The most common variety is a decorative polished brass mechanism under a glass dome.  You may have seen one.  The balance wheel in an old-fashioned mechanical watch is a kind of torsion pendulum, though the restoring force is provided by a flat coiled spring rather than a long twisted wire.  Torsion pendulums can be made very accurate and have been used in numerous precision experiments in physics.  In this session, you will use one to study simple harmonic motion (SHM) because the restoring torque is very nearly proportional to the angular displacement over quite large angles of twist so that the motion is close to being ideal SHM.  

When the torsion pendulum disk is twisted away from equilibrium by an angle ? the twisted wire exerts a restoring torque proportional to that angle

 

W NT                                                           (1)

 

If the wire is thick and made of stiff material, ? is large.  If the wire is long, ? is small.  The rotational analog of Newton’s Second Law is W I₀D, where D d²T/dt²  is the angular acceleration.  So we obtain an equation in the simple harmonic form:

 

d2 T

I_{0   2} NT                                                           (2) dt

 

 

The solution to this equation is:

 

T(t) Acos(Zt I)                                                           (3)

 

so the disk oscillates with time according to this equation.  Here, A is the amplitude of the oscillation and Z is the angular frequency.   The period of the motion (the time for one cycle) is T 2S/Z.  The angular velocity and acceleration of the torsion pendulum are given by:

dT

 AZsin(Zt I)                                                           (4) dt

 

 

d2 T        2

2  AZ cos(Zt I)                                                       (5) dt

 

 

 

so the maximum (absolute) value of the angular velocity is AZ and the maximum (absolute) value of the angular acceleration is AZ².  

 

The angular frequency of oscillation is determined by the physical parameters of the torsion pendulum:

Z

0

I

N

                                                           (6)

 

 

So the time for a single oscillation of the torsion pendulum is the period:

 

2S       I ⁰

T         2S                                                           (7)

Z        N

             

Reading Notes 

               

Name ____________________________ CRN ____________________________ Lab 1: Torsion Pendulum PHYS 2306 Pre-lab Assignment Question Pre-1:  The function

Date ____________________________ GTA Name ______________________ Graph 1. 1

shown in Graph 1. 1 has the form given in Equation (3) of the introduction to this lab.  From the graph, estimate the value of the amplitude A and the angular frequency Z.

 

Question Pre-2:  The angular position of a torsion pendulum as a function of time is given by T(t) Acos(Zt I) . 

1. What do you expect the maximum (absolute) value of the angular velocity dT/dt to be?

Show your work.  Write your answer here and in Prediction 1.1 for reference during the lab.

 

2. What do you expect the maximum value of the angular acceleration d²T/dt² to be?  Show your work.  Write your answer here and in Prediction 1.2 for reference during the lab.

Figure 1. 2

Question Pre-3:  A reference ring with a moment of inertia I_C about the

axis rests on a solid disk torsion pendulum with a moment of inertia I₀, as shown in Figure 1. 2.

1. What is the period T₀ for motion with the disk alone?  Assume that the torsion constant of the wire is N.  (See Equation (7) in the introduction.)
2. What is the period of motion, T₁, for the combined system (i.e., when the reference ring is placed on top of the disk, and both disks are rotating with each other)?  Assume that the torsion constant of the wire is N.  (See Equation (7) in the introduction.)
3. Use the expressions from a) and b) to derive an expression for I₀ in terms of T₀, T₁, and I_C. 

Write your answer here and by Question 2.3 for reference during the lab.

Each Capstone file typically contains a video of an experiment. At the bottom of the Capstone window are controls for that video: 

 

 

one frame at a time.

experiment.

x     Jumps the experiment to the beginning (left arrow) or the end (right arrow) x      Moves the experiment backward (left arrows) or forward (right arrows)

 

x     Plays the experiment at the chosen playback speed. It also pauses the

x     Causes the experiment to repeat when it finishes.

x     Allows you to control the playback speed of the experiment x  This Experiment Selector drop-down menu allows you to switch between experiments if multiple experiments are stored under a single tab. 

 

 

 

 

 

This lab contains post-lab questions. When 10 minutes are remaining in the lab session, stop what you are doing, and answer the post-lab questions.

1. Simple Harmonic Motion

1.1 Preparing your lab station

From the desktop of the computer, open the Class Notes folder and open the Lab 1.cap file.

If needed, clean up your workstation and untangle any cords.

 

1.2 Torsion Pendulum

The torsion pendulum that you will use in this lab is shown in Figure 1. 3.  An aluminum platform sits on top of a  rotary motion sensor (RMS), and the RMS is attached to a vertical wire that is under tension. When you rotate and hold the aluminum platform, the wire becomes twisted.  When you release the platform, the wire untwists, and the torsion pendulum oscillates back and forth. 

 

Figure 1. 3

You can use three different wires for this torsion pendulum.  The wires have the same length, but varying thicknesses.  We will refer to these wires as thin, medium, and thick wires.  

Install the thin wire by attaching it at both the upper and lower wire clamp.  First, attach the top of the wire to the upper wire clamp. Then, have one person hold the upper wire clamp in place while another person adjusts the lower wire clamp and attaches the bottom of the wire to the lower clamp. 

If the round black ring is sitting on top of the torsion pendulum platform, remove it.  The aluminum platform should be secured in place by a bolt.  

 

Open Capstone File - Experiments and Data - Lab 01 - 2306 Part 1.cap.

1.3 Activity: Simple Harmonic Motion

In this activity, you will measure the rotational motion of the torsion pendulum and determine if your measurements are consistent with simple harmonic motion. 

Procedure 1.3.1

In Capstone, navigate to the tab named Procedure 1.3.1. Under this tab are time-series graphs for the angular position, velocity, and acceleration of the torsion pendulum.

Click Play  in the video controls to start the video and begin recording data. In this experiment, the platform is Initially at rest. We then rotate the aluminum platform, release it, and the platform starts oscillating. 

Screenshot

Take a screenshot of your data using the Screenshot Tool           , which adds the screenshot to the journal in Capstone. Open the journal by using the Journal Tool      . Save your screenshot as a jpg or PDF, and include it in your assignment submission. 

Automatic Annotations on Printouts

To save paper, the printers in the lab room automatically annotate printed pages with your quadrant and table number at the bottom of the page.  The annotations have the following format:

NB103-Q(quadrant letter)(table number) (time)

For example, the annotation “NB103-QA3 16:30:49” means that the printed page came from Table 3 in Quadrant A at 16:30 hours.  When retrieving your printouts, check the annotations on all the printouts at the printer station.  After two minutes, if you cannot find your printouts, talk to your TA before executing another print job.

If you need to dispose of a printout, please put it in the Reuse Pile. We use this paper for scratch work in office hours. 

 

 

 

Question 1.1: How do would you determine the period and amplitude from angular position data?

 

 

 

Coordinate Tool

In Question 1.2 below, you will need to use the Coordinate Tool  to extract information from the time-series graphs.  If you are unfamiliar with this tool, read the operation instructions below.

How To Operate the Coordinate Tool x Left-click on the  button in the top menu bar. Square crosshairs will appear on the graph.

x    Left-click on the crosshairs and drag it to the point of interest on the plotted data. The coordinates of this point of interest will be displayed beside the crosshairs.

x    The Coordinate Tool can also be used to measure the difference in horizontal or vertical coordinates of two points. For instance, say you wanted to measure the time difference between two points. Let’s call them Point 1 and Point 2. To do so, follow the steps below:

• Drag the crosshairs of the Coordinate Tool to Point 1
• Right-click on the crosshairs. A drop-down menu will appear.  Click on Show Delta Tool. A shaded box will appear along with another square crosshair attached to a corner of this shaded box.
• Drag the second square crosshair to Point 2. A value will appear on the graph, ùt, which indicates the difference in time between Point 1 and Point 2.

 

Question 1.2: Use the Coordinate Tool to determine the amplitude and the period of the torsion pendulum from the angular position data.  From the period, determine the angular frequency.

Amplitude = ______________________

Period = __________________________

Angular frequency = Z = __________________________

               

Prediction 1.1:  Using the quantities measured in Question 1.2, predict the maximum angular velocity (dT/dt) of the torsion pendulum.  (Use your answer to Question Pre-2a.)

Question 1.3: Use the Coordinate Tool to measure the maximum angular velocity of the pendulum. (Alternatively, you may use the Highlighter     + Statistics  Tools  to perform this measurement.) Record that value below.

Are the measurements consistent with your predictions in Prediction 1.1? If there are discrepancies, discuss those discrepancies. Was there anything wrong with your assumptions or reasoning in Prediction 1.1?

Prediction 1.2: Predict the maximum angular acceleration (d²T/dt ²) of the torsion pendulum. 

(Recall the answer to Question Pre-2b).

Question 1.4: Use the graph tools to measure the maximum angular acceleration of the pendulum. Record that value below.

Are your measurements consistent with your predictions in Prediction 1.2? If there are discrepancies, discuss those discrepancies. Was there anything wrong with your assumptions or reasoning in Prediction 1.2?

 

 

 

1.4 Activity:  The Relationship Between Period and Amplitude for Simple Harmonic Motion

For this activity, you will study how the period of simple harmonic motion depends on the amplitude of the motion.  

Prediction 1.3:   Consider two data runs with the torsion pendulum.  Assume that one run has a larger amplitude than the other.  Does the run with the larger amplitude have a period that is greater, the same, or less than the other run?  Explain your answer.

 

 

 

 

Test your prediction

Test your prediction in Prediction 1.3. 

Navigate to the tab named Prediction 1.3. In this experiment, we oscillated the pendulum with three different initial amplitudes with the thin wire installed. 

Use the Experiment Selector  in the video controls to choose a data run.

(Alternatively, you can use the tab Prediction 1.3: Data view the data from all three data runs simultaneously.) 

Use the available graph tools to obtain the amplitude and period for each run and record them in Question 1.5

Question 1.5: Use the graph tools to measure the amplitude and period for each run and record them below:

Run 1:   Amplitude = _____________________     Period = _______________________

Run 2:   Amplitude = _____________________     Period = _______________________

Run 3:   Amplitude = _____________________     Period = _______________________

Use the Data Selection Tool to display all three data runs simultaneously.

Screenshot

Take a screenshot of your data using the Screenshot Tool           , which adds the screenshot to the journal in Capstone. Open the journal by using the Journal Tool      . Save your screenshot as a jpg or PDF, and include it in your assignment submission. 

Question 1.6:   From the data, does the period depend on the amplitude for simple harmonic motion?  If so, how?

Is your data consistent with your predictions in Prediction 1.3? If there are discrepancies, discuss those discrepancies and their sources. 

 

 

 

 

Open Capstone File - Experiments and Data - Lab 01 - 2306 Part 2.cap.

2.  Angular Frequency of Torsion Pendulum

In this section, you will measure the torsion constant and moment of inertia of the pendulum to predict the angular frequency of the pendulum.  You will then see how that prediction compares to the measured angular frequency.

2.1 Activity: Determination of the Platform’s Moment of Inertia

You first need to determine the moment of inertia of the torsion pendulum, which is mostly due to the aluminum platform and the rotating part of the rotary motion sensor. You could estimate this moment of inertia by disassembling the pendulum, weighing each constituent part, and measuring the radii of each part.  However, you will use another, less destructive method.  You will use a ring with a known moment of inertia to indirectly determine the moment of inertia of the platform. (This is the same method that you used in Question Pre-3. )

Procedure 2.1.1

In Capstone, navigate to the tab Procedure 2.1.1 to measure the necessary quantities to determine the moment of inertia of the black ring, which is called the reference ring. Using the Magnifier Tool, you can use your cursor to zoom in on the image. 

If you need help reading the caliper, open the file [Windows] Capstone File – How To Use a Caliper.cap.

Measure the following quantities:

Mass of reference ring = 0.46778 kg

Inner radius of reference ring = a  = __________________________________________

Outer radius of reference ring = b = __________________________________________

Question 2.1:   From the measurements, calculate the moment of inertia of the reference ring. 

The moment of inertia of a ring of inner radius ?_in, outer radius ?_out, and mass  about an axis perpendicular to it running through its center is

 

 

 

 

 

 

 

Procedure 2.1.2

For this experiment, we installed the thin wire in the torsion pendulum.  

In Capstone, continue using the tab named Torsion 1.

Question 2.2: Measure the period of the torsion pendulum with just the aluminum platform installed. We will call this period T₀.

In the space below, write down the period obtained in Question 1-5.

 

T₀  = __________________________________________

             

Next, install the reference ring on top of the rotating platform.  Measure the period of the combined system, which we will call T₁.   

Navigate to the tab named Procedure 2.1.2_, where we have installed the reference ring on top of the rotating platform. Measure the period of the combined system, which we call T₁.

 

T₁   = ___________________________________________

 

Question 2.3:   From the data, determine the moment of inertia of the torsion pendulum with just the aluminum platform installed.  Use the expression you derived for Question Pre-3.

 

             

Open Capstone File - Experiments and Data - Lab 01 - 2306 Part 3.cap.

 

2.2 Activity: Determination of the Torsion Constant

In this activity, measure the torsion constant and the oscillation period for the three different wires.  

Procedure 2.2.1

Leave the thin wire installed in the torsion pendulum.  

For the upcoming experiments in this section, we removed the black reference ring from the platform.

To measure the torsion constant, we will apply a torque to the system using a string wrapped around the Large Pulley on the rotary motion sensor.  You will determine the magnitude of the applied torque using readings from a force sensor.  By applying varying torques, you can plot the angular displacement of the torsion pendulum versus the applied force; the slope of this line is related to the torsion constant N.The setup is shown in Figure 1. 4.

 

 

 

 

Figure 1. 4

               

Question 2.4:   In Capstone, plot the angular displacement of the torsion pendulum versus the pulling force of the string. Capstone will then apply a linear fit to the plot.  From the slope of the linear fit, how will you determine the torsion constant N?  

Write an expression that relates the torsion constant to the slope. Your expression should include the radius of the Large Pulley on the RMS. (Hint: Use Equation (1) in the introduction.)

 

 

 

 

 

 

Measure the Radius of Large Pulley on RMS

Navigate to the tab Pulley Measurement. Use the Vernier caliper to measure the diameter of the Large Pulley on the RMS  ???_RMS. 

Question 2.5: Determine the radius of the pulley from the measured diameter and record the value of the radius here:  

?_RMS = ______________________________________

 

 

 

Replace the aluminum platform.

For each of the three wires, you will measure (1) its oscillation period and (2) its angular displacement in response to a torque. You will then record these values in Question 2.6 For these experiments, each wire has its own Capstone file:

x    Thin Wire – Capstone File - Experiments and Data - Lab 01 - 2306 Part 3.cap. x           Medium Wire–Capstone File - Experiments and Data - Lab 01 - 2306 Part 4.cap. x            Thick Wire – Capstone File - Experiments and Data - Lab 01 - 2306 Part 5.cap.

             

Measuring the Oscillation Period

To measure the oscillation period of a given wire, follow the steps below:

x    In Capstone, navigate to the tab named Oscillation Period. x            Start recording data.

x    Rotate the aluminum platform of the torsion pendulum platform and release it.  The torsion pendulum will begin oscillating.

x    Capstone will begin plotting the angular displacement of the pendulum vs. time. From this graph, determine the oscillation period of the pendulum. Record this value in Question 2-6.

Measuring Angular Displacement vs. Force

To measure the angular displacement of a given wire vs. the pulling force of the string, follow the steps below.

x    In Capstone, navigate to the tab named Theta vs. F. x           Remove the aluminum platform on top of the pulley.

x    Keeping the pulley stationary, wrap the string around the largest pulley 2-3 full turns in the direction indicated in Figure 1. 4.

x    Attach the end of the string to the hook of the force sensor, as shown in Figure 1. 4.

x    Replace the aluminum platform x       Start recording data and then gently pull on the force probe.

x    Capstone will begin plotting the angular displacement of the wire vs. the pulling force, and Capstone will automatically perform a linear fit of this plot.   Record the slope in Question 2.6.

 

 

 

Question 2.6: Measure the following quantities using the procedures detailed in Measuring the Oscillation Period and Measuring Angular Displacement vs. Force subsections.

Thin wire:

Oscillation period = T₁ = ___________________________________________________ Slope of angular displacement versus applied force =   _______________________ Medium wire:

Oscillation period = T₂ = ___________________________________________________

Slope of angular displacement versus applied force =   _______________________

Thick wire:

Oscillation period = T₃ = ___________________________________________________

Slope of angular displacement versus applied force =   _______________________

Question 2.7:   From the measured slopes in Question 2.6, determine the torsion constant for each of the three wires. Write down the formula that you used to compute these values.

Thin wire:             N₁ = _______________________________

Medium wire:     N₂= _______________________________

Thick wire:           N₃ = _______________________________

 

 

Prediction 2.1:   From your measured torsion constants and moment of inertia, write down the predicted angular frequency and period for each of the three wires.  Write down the formulae that you used for your computations.

Thin wire:              Z₁ = ___________________     T₁ = ____________________

Medium wire:     Z₂= ____________________    T₂ = ____________________ Thick wire:            Z₃ = ___________________    T₃ = ____________________

 

 

 

Question 2.8:   Summarize your predicted and measured periods for each of the three wires below.  Determine the percentage difference = (100)* |(predicted - measured)|/predicted.

 

Thin wire:           T_pred = ____________    T_meas = ___________    % diff = ____________

Medium wire:   T_pred = ____________    T_meas = ___________    % diff = ____________

Thick wire:         T_pred = ____________    T_meas = ___________    % diff = ____________

 

               

Question 2.9: Is Equation (6) in the introduction a good model for the data? Explain. 

3.Clean up

Out of respect for the students and TAs that will use this lab station after you, please clean up your lab station. 

x           Disconnect any wire connected to the torsion pendulum x      Un-tangle and organize wires and strings.

x           Collect small items and put them back in their containers or bags. x Neatly organize the remaining items at your station.      

Lab 1 Post-Lab Questions

 

Question Post-1: You observe an oscillating torsion pendulum, and you determine that its angular displacement is well-described by the following equation:

?(?)= ????(??)

where   = 0.5 radians is the amplitude and Z = 1.0 s^-1 is the angular frequency.  What is the maximum angular speed of the torsion pendulum during its motion?

Question Post-2: When a torsion pendulum oscillates, does the period of motion depend on the amplitude?  Explain.