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Large Amplitude Pendulum EX-9905 Page 4 of 4


Large Amplitude Pendulum


EQUIPMENT

INCLUDED:

1

Large Rod Stand

ME-8735

1

45 cm Long Steel Rod

ME-8736

1

Pendulum Accessory

003-05971

1

Rotary Motion Sensor

CI-6538

NOT INCLUDED, BUT REQUIRED:

1

500 Interface

CI-6400

1

DataStudio

CI-6870


INTRODUCTION


This experiment explores the dependence of the period of a simple pendulum on the amplitude of the oscillation. Also, the displacement, velocity, and acceleration for large amplitude are plotted versus time to show the difference from the sinusoidal motion of low amplitude oscillations.


A rigid pendulum consists of a 35-cm long lightweight (28 g) aluminum tube with a 75-g mass on each end, with the center of the tube mounted on a Rotary Motion Sensor. One of the masses is slightly closer to the center than the other mass, so the pendulum will oscillate slowly to allow time to view the motion of the pendulum while also watching the real-time graph of displacement, velocity, and acceleration versus time.


The period is measured as a function of the amplitude of the pendulum and compared to theory.



THEORY


A simple pendulum consists of a point mass at a distance L away from a pivot point. In this experiment, a mass is attached to a lightweight rod and the mass is concentrated enough to assume it is a point mass and the rod's mass can be neglected.


The period of a physical pendulum is given by

LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE (1)


for small amplitude (less than 20o). I is the rotational inertia of the pendulum about the pivot point, m is the total mass of the pendulum, and d is the distance from the pivot to the center of mass.

For larger amplitudes, the restoring torque is not linear and the period is given by an infinite series:


LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE (2)

LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE

where n is an integer and α is the amplitude (angle). The first five terms are given by

Equation (3):

LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE

LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE

To slow the oscillation, two masses are used with one mass slightly closer to the pivot than the other mass.


The component of gravity that is tangent to the circular path of the pendulum bob is shown for several angles in Figure 1.


LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE
















Figure 1: Tangential Acceleration Figure 2: Pendulum Setup


SET UP


1. Put the Rotary Motion Sensor on the rod stand and plug it into Channels 1 and 2 on the ScienceWorkshop 500 interface. See Figure 2.



2. Put the large step of the pulley outward on the Rotary Motion Sensor and attach the pendulum rod at its center. Attach the two brass masses on the ends of the rod, with one at the end and the other about 0.5 cm from the other end.


3. Open the file called "1Large Amp Pend".


PROCEDURE


Small Amplitude


1. Click on START with the pendulum at rest in its equilibrium position. This will set the angle on the Rotary Motion Sensor to zero at the equilibrium position.


2. Displace the pendulum less than 20o from equilibrium. Let it go and click STOP after a few oscillations.


3. Examine the graphs of angular displacement, angular velocity, and angular acceleration.


A. Are they sinusoidal?

B. Which graphs are in phase with each other (i.e. their maxima coincide).

C. Are the periods the same?

D. Measure the period using the Smart Tool at the top of the graph.

E. Derive an expression for the theoretical period of this physical pendulum for small amplitudes using Equation (1). Measure the masses and lengths and calculate the theoretical period. Compare it to the measured period.


Large Amplitude


1. Click on START with the pendulum at rest in its equilibrium position. This will set the angle on the Rotary Motion Sensor to zero at the equilibrium position.


2. Displace the pendulum nearly 180o from equilibrium. Let it go and click STOP after a few oscillations.


3. Examine the graphs of angular displacement, angular velocity, and angular acceleration. Click on a Smart Cursor for each of the three graphs to align the velocities and accelerations with various angles (180o, 90o, and 0o).

A. Are they sinusoidal?

B. Which graphs are in phase with each other (i.e. their maxima coincide).

C. Are the periods the same?

D. Measure the period using the Smart Tool at the top of the graph of the angular displacement.

E. Is the period longer or shorter than the low amplitude period?

4. For the angular acceleration graph, identify the angles at which the major features of the acceleration curve occur. Explain what forces cause the acceleration graph to look like it does. What would the acceleration graph look like if the pendulum was released from rest at 179.9o?


Period vs. Amplitude


1. Open the file called "2Large Amp Pend". With the pendulum at rest at its equilibrium, click on START and displace the pendulum nearly 180o from equilibrium and let it go. Let the pendulum oscillate until the amplitude is less than 5o. Then click on STOP.


2. Examine the graph of Period vs. Amplitude.


A. At what angle is the period the longest?

B. Does the period at low amplitude match the period measured before in the Low Amplitude portion of the lab?

C. If the pendulum was released from exactly 180o, what would the period be?


3. Use the DataStudio calculator to calculate the function (see Equations 2 and 3) that approximates the period for all amplitudes. Graph this function on the same graph as the Period vs. Amplitude data. Do they match?


Written by Ann Hanks LARGE AMPLITUDE PENDULUM EX9905 PAGE 4 OF 4 LARGE


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