The University of Hong Kong

Department of Physics

Physics Laboratory

PHYS3350 Classical Mechanics

Experiment No. 3350-2

The Physical Pendulum

1.1 Aim of the lab

This lab allows students to meet several goals:

1. Study motion of physical pendulum

2. Measure the value of gravitational acceleration g

3. Learn basics of data analysis

4. Learn basics of work with PASCO data acquisition system.

1.2 Introduction

Physical pendulums have many applications in industry and science. One of the most known ap-plications of a pendulum is pendulum clock, the first one was build in 1600s. In 1851 the Foucaultpendulum was introduced and served as a first direct experiment to measure the rotation of Earth.Pendulums are widely used to measure gravitation acceleration g, and thus to study the geologyand search for minerals. Pendulums are also used in seismometers to measure the magnitude ofearthquakes and many have many other applications.

This laboratory work is focused on the physics on physical pendulum. Students will learn basicproperties of physical pendulum and measure the gravitational acceleration g.

1.3 Theoretical background

Mathematical (simple) pendulum is a system, which consists of a massive object with negligiblesize, and a massless, long string. Simple pendulum is usually used as a model and simplification ofrealistic objects (physical pendulum).

The Simple PendulumOne should describe mathematical pendulum using Newton’s equations:∑

~Fi = m~a (for translational motion) (1.1)

d~L

dt= ~τ ⇔ I

d~ω

dt= ~r ×

∑~Fi (for rotational motion) (1.2)

Schematic view of the simple pendulum can be found on Fig. 1.1. Two forces are acting on an

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Figure 1.1: Schematic view of a simple pendulum

object during the movement: gravitation force and tension of a string. From equation (1.1), it iseasy to get equation of motion for simple pendulum:

d2θ(t)

dt2+g

lsin θ(t) = 0 (1.3)

Equation (1.3) does not have exact solution in general case. In case, when oscillations occur nearthe point of equilibrium (θ 1), the period of oscillations T equals to:

T =2π

ω= 2π

√l

g(1.4)

Physical Pendulum: oscillations on small angleTo find the equation of motion for physical pendulum, one should take rotational effects into account:

d~L

dt= ~τ ⇔ I

d~ω

dt= ~r ×m~g (1.5)

Angular velocity ~ω equals to: ~ω = d~θdt

. Since, angular velocity ~ω has only one component, it is easyto write equation (1.5) in scalar form:

I · d2θ(t)

dt2= −lmg · sin θ(t)⇔ d2θ(t)

dt2+ ω2 · sin θ(t) = 0 (1.6)

where ω =√

lmgI

. In case of small angle of oscillations, equation (1.6) is simplified:

d2θ(t)

dt2+ ω2θ(t) = 0⇒ θ(t) = A · cos(ωt) +B · sin(ωt) (1.7)

As one can see, physical pendulum follows the same equation of motion as a simple pendulum. Theonly difference is the period of oscillations. Period of oscillations is derived in the following way(using Parallel Axis Theorem):

T =2π

ω= 2π

√I

mlg= 2π

√ICM +ml2

mlg= 2π

√ICMmlg

+l

g(1.8)

2

Figure 1.2: Schematic view of a physical pendu-lum

Figure 1.3: Example of coupled oscillations

where

ICM =1

12m(L2 +B2) - moment of inertia of a rectangular rod, L is its length and B - thickness;

l - distance from the pivot to the center of mass;

g = 9.8m/s2 - acceleration of a free fall;

m - mass of the rod.

It is easy to show from equation (1.8), that gravitational acceleration equals to

g = 4π2(ICM +ml2

T 2 ·ml

)(1.9)

Physical Pendulum with Energy DissipationOne additional force must be taken into account to describe the experimental data correctly -friction of the air. Air resistance will cause dissipation of kinetic energy from pendulum. In firstapproximation, air resistance is proportional to the velocity of the moving object. Differentialequation will have the following form (case of small oscillations):

d2θ(t)

dt2+ 2β

dθ(t)

dt+ ω2

0θ(t) = 0 (1.10)

where β is attenuation coefficient. It is easy to show, that solution of (1.10) can be presented in thefollowing form:

θ(t) = A · e−βt · sin(ωt+ φ) (1.11)

where ω =√ω2

0 − β2. Equation (1.11) describes damping oscillations, and presented on Fig. 1.4. Itis easy to show from equation (1.11), that gravitational acceleration g equals to:

g = (ω2 + β2)ICM +ml2

ml↔ g = (

4π2

T 2+ β2)

ICM +ml2

ml(1.12)

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Figure 1.4: Example of damping oscillations

Coupled Physical PendulumsThe example of coupled oscillations is shown in Fig. 1.3. Coupled oscillations occur, when twopendulums are connected by a spring. This connection causes an elastic force (F = k∆x), actingon each pendulum. Such system is described by the following set of equations, in case of smalloscillations:

θ1 + ω21θ1 + kl2

I(θ2 − θ1) = 0

θ2 + ω21θ2 − kl2

I(θ2 − θ1) = 0

(1.13)

where ω1 =√

lmgI

. After some transformations, it is easy to obtain the following set of differential

equations: (θ1 + θ2) + ω2

1(θ1 + θ2) = 0

(θ1 − θ2) + ω22(θ1 − θ2) = 0

(1.14)

where ω2 =√

lmgI

+ 2kl2

I. It is easy to show, that system (1.14) has the following solutions:

θ1 = 12A1 sin(ω1t+ φ1) + 1

2A2 sin(ω2t+ φ2)

θ2 = 12A1 sin(ω1t+ φ1)− 1

2A2 sin(ω2t+ φ2)

(1.15)

Solutions (1.15) can be simplified, by choosing initial time such, that φ1 = φ2 = 0. Amplitudes ofoscillations can be set the same (A1 = A2):

θ1 = A cos(ω1−ω2

2t) sin(ω1+ω2

2t) = A cos(Ωt) sin(ωt)

θ2 = A sin(ω1−ω2

2t) cos(ω1+ω2

2t) = A sin(Ωt) cos(ωt)

(1.16)

where Ω = ω1−ω2

2, ω = ω1+ω2

2. Example of coupled oscillations is shown in Fig. 1.5. Two oscillations

have phase shift of π2, as it can be seen from equation (1.16). During such coupled movement, the

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Figure 1.5: Example of damping oscillations

energy from one pendulum will be transferred to another pendulum, and vice versa. The period ofsuch energy transfer TΩ equals to:

TΩ =2π

Ω=

2π

|ω1 − ω2|=

1∣∣∣ 1T1− 1

T2

∣∣∣ (1.17)

One should note, that system of equations (1.16) does not include damping effected, as the amplitudeis not decreasing with time. Damping effects can be taken into account in the same way, as it wasdone in equation (1.10).Pendulum Oscillations: case for big angleEquation (1.7) describes oscillations near the point of equilibrium, but it does not work well in thecase of large angles. Unfortunately, equation (1.7) does not have analytical solution, but motionfor physical pendulum for large angles can be solves with some approximations. We will considercase of simple pendulum at first. Suppose, that the motion starts at initial angle θin. The law ofconservation of energy gives us:

Ein = Ef ⇐⇒ mgl(1− cos θin) =mv2

2+mgl(1− cos θ(t)) (1.18)

It is easy to see, that during circular motion v = l · dθdt

. Equation (1.18) can be rewritten as thefollowing:

mgl(1− cos θin) =ml2

2

(dθdt

)2

+mgl(1− cos θ(t))→ dθ

dt=

√2g

l

√cos θ(t)− cos θin (1.19)

One can present equation (1.19 with separated variables. After some transformations, it can berewritten as the following:

dθ√sin2(θin/2)− sin2(θ(t)/2)

= 2

√g

ldt (1.20)

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It is proposed to make some substitutions for simplicity. Let sin θi/2 = b (|b|≤1), andsin2(θ(t)/2)

sin2(θin/2)= sinφ. Equation (1.20) takes such shape:√

g

l

∫ t

0

dt =

∫ φ(t)

0

dφ√1− b2 sin2 φ

(1.21)

Function of the right side of equation (1.21) is called Jacobin elliptical function. After the integrationover the period of motion T and expanding function inside the integral, one have the following:√

g

lT =

∫ 2π

0

dφ[1 +1

2b2 sin2 φ+

1

8b4 sin4 φ+ ...] = 2π(1 +

1

4b2 +

9

64b4 + ...) (1.22)

From equation (1.22) it is easy to see, that:

T = 2π

√l

g(1 +

1

4sin2(θi/2) +

9

64sin4(θi/2) + ...) (1.23)

As one can see, in the limit θin → 0 equation (1.23) transforms in the well known formula for simpleoscillations (1.4). For physical pendulum we have the following formula:

T = 2π

√I

mgl(1 +

1

4sin2(θi/2) +

9

64sin4(θi/2) + ...) (1.24)

The reader is recommended to write the formula (1.24) for physical pendulum in the next leadingorder (sin6(θin/2)).

1.4 Experimental Setup and Procedure

Experimental equipment consists from following parts:

1. 1 × PASCO CI-6538 Rotary Motion Sensor

2. 1 × PASCO CI-6691 Mini-Rotational Accessory

3. 1 × PASCO ME-9348 Mass and Hanger Set

4. 1 × PASCO ME-9355 Base and Support Rod

5. Beam balance

6. Caliper

1.4.1 Experiment 1: Minimum period of a bar

PurposeThe purpose of this experiment is to calculate lmin, the distance from the pivot of a given barof length L to the point of center of gravity that would give the minimum period of oscillation forthe bar. Measure the distance that gives the minimum period and compare it to the calculated result.

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Experimental procedure

1. Measure and record the length, L, of the bar. Mount the Rotary Motion Sensor on a supportrod so that the shaft of the sensor is horizontal (parallel to the table).

2. Use a mounting screw to attach the bar to the shaft of the sensor through the first hole abovethe center hole of the bar. In other words, attach the bar so the pivot point is 2 cm above thecenter of gravity (see Fig. 1.6 for reference).

3. Connect the sensor to a PASCO interface and connect the interface to a computer.

4. On the computer, start the DataStudio program. Set up the program so that it has a Graphdisplay of Angular Position (rad) versus Time (s) (see Fig. 1.7 for reference).

5. Gently start the pendulum bar swinging with a small amplitude (about 10-20 degrees total).

6. Click “Start” to begin recording data. After about 25 seconds, click “Stop” to end recordingdata. Data will appear in the graph of angular position versus time and also in the graph ofperiod versus time.

7. Move the mounting screw to the next hole (4 cm from the center hole).

8. Start the pendulum bar swinging and record data for about 25 seconds.

9. Repeat the process for the holes that are 6 cm, 8 cm, 10 cm, 12 cm, and 14 cm from the centerhole.

10. Save the results as a file “UID exp1.ds” where UID is your University no. (e.g. 2011123456 exp1.ds)

11. Repeat steps 5-11 for two more trials.

Figure 1.6: Experimental setup: rect-angular rod

Figure 1.7: Graph display

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Data Section

Length of pendulum bar, L:Calculated value for length that gives minimum period (∂T

∂l= 0⇒ lmin = 1√

12L):

Measured value for length that gives minimum period:Percent difference:

Data Analysis

1. Create a Table display to show period versus length. In the “Experiment” menu, select “NewEmpty Data Table”.

2. Double click the label of the new Table display in the Summary panel to open the DataProperties window. Give the table a Measurement Name of “Period versus Length”, an “X”Variable Name of “Length” with “cm” for units, and a “Y” Variable Name of “Period” with“s” for units.

3. Find the period of oscillation for the 2 cm setup.

(a) Click the “Smart Cursor” button in the toolbar.

(b) Move the Smart Cursor to one of the first peaks of Angular Position.

(c) Hover the cursor over the Smart Cursor until the “delta” symbol appears.

(d) Click and drag the “delta” symbol to the tenth peak of Angular Position.

(e) Divide the time for ten oscillations by ten and record the number as the period of oscil-lation.

4. In the Table display, enter “2” as the first length in the “x” column and the period of oscillationfor the 2 cm length as the first period in the “y” column. Continue to enter data points in theTable display.

5. Click and drag a Graph display icon from the “Displays” part of the Summary panel to the“Data” under “Period versus Length” in the top part of the Summary panel. The Graphdisplay opens with “Period” on the Y-axis and “Length” on the X-axis.

6. Determine which length gives the minimum period of oscillation of the pendulum bar andrecord this length in the data section.

Questions

1. What is the percent difference between the calculated value for the length that gives minimumperiod of oscillation and the measured value for the length?

2. Derivation of the length for minimum period, lmin = 1√12L

(Hint: Take the derivative of the expression (1.8) for the period of oscillation and find conditionon minimum value of period T.)

3. Would two pendulum bars with different masses but with the same dimensions have a differentvalue for the length, that gives minimum period of oscillation? Why or why not?

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Figure 1.8: Angular position as a function oftime

Figure 1.9: Example of the data table

4. Derive equation (1.3).

5. Amplitude of oscillations decrease with time. Why?

6. Fit the distribution of θ(t), obtained experimentally, with function (1.11). From the fit estimatethe value of attenuation parameter β, and frequency ω. Estimate the value of period ofoscillations T from data. Compare T estimated from data directly with T = 2π/ω.

7. Perform error analysis to all calculations.

1.4.2 Experiment 2: Use a Physical Pendulum to Measure the Accel-eration Due to Gravity, g

PurposeThe purpose of this experiment is to use a physical pendulum to measure the acceleration due togravity.

Experimental procedures

1. Measure and record the length, L, and width, B, of the bar.

2. Measure and record the mass, m, of the bar.

3. Mount the Rotary Motion Sensor on a support rod so that the shaft of the sensor is horizontal(parallel to the table).

4. Use a mounting screw to attach the bar to the shaft of the sensor through the hole at the endof the bar. In other words, attach the bar so the pivot point is at the very end of the bar.

5. Start the Data Studio program. Set up the program so that it has a Graph display of AngularPosition (rad) versus Time (s).

6. Open the Calculator and select “period(10,10,1,x)” from the “Special” menu in the Calculator.[This function determines the period of oscillation from the angular position versus time data.]

7. To define the variable “x” in the Calculator, click the menu down arrow under “Variables” andselect “Data Measurement”. Select “Angular Position” from the window that opens.

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8. Click “Properties” in the Calculator window. Give the function “Period” as the measurementname and the variable name. Enter “sec” (seconds) as the unit.

9. Create a graph of “Period Versus Time”. Click and drag the Graph icon from the Displays listin the Summary panel to the period function in the Data section of the Summary panel.

10. In the “Statistics” menu of the “Period Versus Time” Graph display, select “Mean”. Click the“Statistics” button in the toolbar so that the legend box will show both the data run and themean.

11. Gently start the pendulum bar swinging with a small amplitude (about 20 degrees total). Click“Start” to begin recording data. After about 30 seconds, click “Stop” to end recording data.Data will appear in the graph of angular position versus time and also in the graph of periodversus time.

12. Save the results as a file “UID exp2.ds” where UID is your University no.(e.g. 2011123456 exp2.ds)

13. Repeat steps 11-13 for two more trials.

14. Repeat the experiment for case of large angle. Set initial value of θin ≈ 70-90o

Figure 1.10: Experimental setup: rectangularrod Figure 1.11: Data Studio program: Calculator

Data SectionLength of pendulum bar, L:Width of pendulum bar, B:Mass of pendulum bar, m:Average period of oscillation, T :Distance from pivot to center of gravity, l:Calculated value for the moment of inertia about the center of gravity, ICM :Calculated value for acceleration due to gravity, g (use formulas (1.9) and (1.12)):Percent difference:

Questions

1. Derive formula (1.9) and (1.12). What approximations did you make? Is this formula general,or it can be used only for limited cases (if so, which cases)?

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2. How does the calculated value for g based on the period compare to the accepted value for g,9.8 ms−2?

3. Do your results confirm that the acceleration due to gravity, g, can be measured accuratelyusing a physical pendulum? Why or why not?

4. Is the amplitude of 10-20 degrees satisfy the approximation small angles? How about 70-90o?Calculate, using equation (1.24), the period T for both cases. Compare your results withexperimental data. What is the percent difference between formula (1.8) and formula (1.24)for small large angles, estimated from data and calculated one?

5. Optional for higher mark. Modify formula (1.9) for the case of high amplitude of oscilla-tions. On what percentage did you results improve with modified formula?

6. Optional for higher mark. Derive equation (1.24) in the next leading order (sin6(θin))).

7. Perform error analysis.

1.4.3 Experiment 3: Coupled pendulums

PurposeThe purpose of this experiment is to study the motion of the coupled pendulums.

Experimental procedures

1. Connect two pendulums with a spring.

2. Connect the sensor to a PASCO interface and connect the interface to a computer.

3. On the computer, start the DataStudio program. Set up the program so that it has a Graphdisplay of Angular Position (rad) versus Time (s) (see Fig. 1.7 for reference).

4. Gently start one pendulum bar swinging with a small amplitude (about 10-20 degrees total).

5. Click Start to begin recording data for each sensor simultaneously. After few minutes ofrecording, click Stop to end recording data. Data will appear in the graph of angular positionversus time and also in the graph of period versus time.

6. Repeat the trial several times.

Questions

1. Fit the distribution of θ1,2(t) for each pendulum with the function (1.16).

2. Extract the value of TΩ and ω from the fit. Compare your results with estimations from thegraph.

3. From the value of TΩ and ω estimate the value of springs constant k.

4. Perform error analysis.

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1.4.4 Experiment 4: Extracting oscillation from videos

PurposeThe purpose of this experiment is to demonstrate how to use handy objects (e.g., cell phones) to dosimple physics experiment. The students will take videos of the pendulum and extract the dampedoscillation from the videos.Experimental procedure

1. Use a mounting screw to attach the bar to the shaft of the sensor through the hole at the top.In other words, attach the bar so the pivot point is 14 cm above the center of gravity.

2. Put a white paper behind the bar as background. If one paper is not enough, take two ormore.

3. Put your cell phone in front of the bar and keep it still and straight up.

4. Open the camera. Adjust the distance properly and prepare for a video recording.

5. Gently start the pendulum bar swinging with a small amplitude.

6. Stand behind the cell phone and take a video for about 30 seconds.

7. Move the mounting screw to two other holes (12 cm and 10 cm from the center of gravity),and repeat the above process.

8. Take more trials if needed.

Video and Data analysisA MATLAB code is available for processing the videos (It can be downloaded at this link, alongwith instructions on how to use it and examples).

1. Copy the videos and the MATLAB code into a computer and put them in the same folder.

2. Open the MATLAB code. Read its instructions and understand what it does.

3. Set videofile_path to be the name of the video file.

4. Set X, Y, cutWidth, cutHight properly so that the background in the resized frames is thewhite paper. You may adjust these parameters multiple times until they are appropriate.

5. Run the code. If the extracted oscillation motion x(t) does not behave nicely, try to adjustthe parameter areacontrol.

6. The oscillation period T and damping coefficient β are given in the outputs. Moreover, it showsthe fitting of the oscillatory motion into Eq. (1.11). Compare T and β with those obtained byPASCO.

7. Repeat the above process for other videos.

Additional study (optional)

1. In the MATLAB code, the period and damping coefficient are obtained by fitting x(t) of thebar bottom. Think about how to obtain them from y(t) of the bar bottom. The extracted x(t)is nicer than y(t), so you may need to improve the algorithm.

2. Study the angular dependence of the period T for large-angle oscillations.

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1.5 Software for data analysis

Information and suggested software for data analysis.Instruction of how to use campus license on MATLAB and Simulink products:

1. Go to HKUs MATLAB Portal.

2. Click “Sign In to Get Started” in the Get MATLAB and Simulink section.

3. You will be prompted to authenticate using your HKU Portal UID and PIN.

4. Create a MathWorks account using your @connect.hku.hk email address.

5. Click the “download” button and choose a supported platform to download and run theinstaller.

6. In the installer, select “Log in with a MathWorks Account” and follow the online instruc-tions.

7. When prompted to do so, select “Academic Total Headcount license labeled Individ-ual”.

8. Select the products you want to download and install.

9. After downloading and installing your products, keep the “Activate MATLAB checkbox” se-lected and click “Next”.

10. Follow the prompts to activate MATLAB.

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Bibliography

[1] PASCO Scientific. (n.d.). Pasco CI-6538 Rotary Motion Sensor Manual. Roseville, CA: PASCOScientific.

[2] Morin, D. (1999). Introduction to Classical Mechanics: With Problems and Solutions. Cam-bridge University Press.

[3] Kleppner D., Kolenkow R. J. (2012). An Introduction to Mechanics. Cambridge UniversityPress.

[4] James F. (2008). Statistical Methods in Experimental Physics.

[5] Leo W. R. (1994). Techniques for Nuclear and Particle Physics Experiments. p. 81

[6] Taylor J. R. (1997). An Introduction to Error Analysis.

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