The Simple Pendulum
by Dr. James E. Parks
Department of Physics and Astronomy
401 Nieln Physics Building
The University of Tenne
Knoxville, Tenne 37996-1200
Copyright June, 2000 by James Edgar Parks*
*All rights are rerved. No part of this publication may be reproduced or transmitted in any form or by
any means, electronic or mechanical, including photocopy, recording, or any information storage or河外星系
retrieval
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The purpos of this experiment are: (1) to study the motion of a simple pendulum, (2) to
study simple harmonic motion, (3) to learn the definitions of period, frequency, and
amplitude, (4) to learn the relationships between the period, frequency, amplitude and
length of a simple pendulum and (5) to determine the acceleration due to gravity using
the theory, results, and analysis of this experiment.
Theory
A simple pendulum may be described ideally as a point mass suspended by a massless
string from some point about which it is allowed to swing back and forth in a place. A
simple pendulum can be approximated by a small metal sphere which has a small radius
and a large mass when compared relatively to the length and mass of the light string from
which it is suspended. If a pendulum is t in motion so that is swings back and forth, its
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motion will be periodic. The time that it takes to make one complete oscillation is
defined as the period T. Another uful quantity ud to describe periodic motion is the
frequency of oscillation. The frequency f of the oscillations is the number of oscillations
that occur per unit time and is the inver of the period, f = 1/T. Similarly, the period is
the inver of the frequency, T = l/f. The maximum distance that the mass is displacedtips是什么意思
from its equilibrium position is defined as the amplitude of the oscillation.
When a simple pendulum is displaced from its equilibrium position, there will be a
restoring force that moves the pendulum back towards its equilibrium position. As the
motion of the pendulum carries it past the equilibrium position, the restoring force
changes its direction so that it is still directed towards the equilibrium position. If the restoring force F G is opposite and directly proportional to the displacement x from the
equilibrium position, so that it satisfies the relationship
F = - k x
G G (1)
then the motion of the pendulum will be simple harmonic motion and its period can be
calculated using the equation for the period of simple harmonic motion
T = 2π
It can be shown that if the amplitude of the motion is kept small, Equation (2) will be
satisfied and the motion of a simple pendulum will be simple harmonic motion, and
Equation (2) can be ud.
θ
Figure 1. Diagram illustrating the restoring force for a simple pendulum.
The restoring force for a simple pendulum is supplied by the vector sum of the
gravitational force on the mass. mg, and the tension in the string, T. The magnitude of
the restoring force depends on the gravitational force and the displacement of the mass
from the equilibrium position. Consider Figure 1 where a mass m is suspended by a
string of length l and is displaced from its equilibrium position by an angle θ and a
distance x along the arc through which the mass moves. The gravitational force can be
resolved into two components, one along the radial direction, away from the point of
suspension, and one along the arc in the direction that the mass moves. The component
of the gravitational force along the arc provides the restoring force F and is given by
F = - mg sin θ (3)
where g is the acceleration of gravity, θ is the angle the pendulum is displaced, and the
minus sign indicates that the force is opposite to the displacement. For small amplitudes
where θ is small, sinθ can be approximated by θ measured in radians so that
Equation (3) can be written as
F = - mg θ. (4)
The angle θ in radians is x
l
, the arc length divided by the length of the pendulum or the
radius of the circle in which the mass moves. The restoring force is then given by国际音标发音
x
F = - mg
l
(5)
and is directly proportional to the displacement x and is in the form of Equation (1) where
mg
k =
l
. Substituting this value of k into Equation (2), the period of a simple pendulum
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can be found by
T = 2π (6) and
T = 2π. (7)
coalTherefore, for small amplitudes the period of a simple pendulum depends only on its
length and the value of the acceleration due to gravity.
Apparatus
The apparatus for this experiment consists of a support stand with a string clamp, a small
spherical ball with a 125 cm length of light string, a meter stick, a vernier caliper, and a
timer. The apparatus is shown in Figure 2.
Procedure
1. The simple pendulum is compod of a small spherical ball suspended by a long, light
string which is attached to a support stand by a string clamp. The string should be
approximately 125 cm long and should be clamped by the string clamp between the
two flat pieces of metal so that the string always pivots about the same point.
Figure 2. Apparatus for simple pendulum.
2. U a vernier caliper to measure the diameter d of the spherical ball and from this
calculate its radius r. Record the values of the diameter and radius in meters.
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3. Prepare an Excel spreadsheet like the example shown in Figure 3. Adjust the length
of the pendulum to about .10 m. The length of the simple pendulum is the distance from the point of suspension to the center of the ball. Measure the length of the string s l from the point of suspension to the top of the ball using a meter stick. Make the following table and record this value for the length of the string. Add the radius of the ball to the string length s l and record that value as the length of the pendulum s l l r =+.
4. Displace the pendulum about 5º from its equilibrium position and let it swing back
and forth. Measure the total time that it takes to make 50 complete oscillations. Record that time in your spreadsheet.
5. Increa the length of the pendulum by about 0.10 m and repeat the measurements
made in the previous steps until the length increas to approximately 1.0 m.
6. Calculate the period of the oscillations for each length by dividing the total time by
the number of oscillations, 50. Record the values in the appropriate column of your data table.
Figure 3. Example of Excel spreadsheet for recording and analyzing data.
7. Graph the period of the pendulum as a function of its length using the chart feature of
Excel. The length of the pendulum is the independent variable and should be plotted on the horizontal axis or abscissa (x axis). The period is the dependent variable and should be plotted on the vertical axis or ordinate (y axis).
8. U the trendline feature to draw a smooth curve that best fits your data. To do this,
from the main menu, choo Chart and then Add Trendline . . . from the dropdown menu. This will bring up a Add Trendline dialog window. From the Trend tab, choo Power from the Trend/Regression type lections. Then click on the Options tab and lect Display equations on chart option.
9. Examine the power function equation that is associated with the trendline. Does it
suggest the relationship between period and length given by Equation (7)?
10. Examine your graph and notice that the change in the period per unit length, the slope
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of the curve, decreas as the length increas. This indicates that the period increas with the length at a rate less than a linear rate. The theory and Equation (7) predict that the period depends on the square root of the length. If both sides of
Equation 7 are squared then
