Oscillations
Introduction
What kind of motion do you see in a playground swing? The motion of a system like this is often
called periodic, oscillatory, or vibratory. The motion is periodic because it repeats itself in equal
intervals of time. It is oscillatory or vibratory because the object moves back and forth over the same
path. You probably also realize that the path grows shorter with time when frictional forces are present
that damp the oscillations.
This laboratory will be devoted to the study of such systems. Before you begin, you should be familiar with
the following definitions and terms:
· A cycle of this motion consists of one complete round trip of the motion measured from any point.
·
The period (symbol T) of the oscillations is the time required to complete one cycle of the motion.
·
The equilibrium position of the system is the position it comes to a long time after it was last disturbed.
Textbook Reading
Physics 150: Halliday, Resnick, and Walker: Chapter 16, especially section 6.
Physics 125: Cutnell & Johnson: Sections 10-1 through 10-4 (pages 248-265).
Overview of Laboratory
In the first part of the laboratory you will use the motion detector to study the way your arms and legs
move, relating them to the textbook “simple” pendulum. You’ll then use the rotary motion sensor to
study the motion of a pendulum in more detail.
A simple pendulum consists of a massive object (or “bob”) on a string or rod that has a
mass very small in comparison to that of the bob. The drawing at the right shows the
geometry of a simple pendulum and the forces on the bob. Textbooks show how
to understand the motion of the pendulum when the angle between the string and
the vertical (equilibrium position) is small enough that sin θ  θ (θ measured in radians). One goal of
this lab is to determine what “small enough” means in practice.
Objectives
·
To become acquainted with both a simple and a physical pendulum.
·
To explore the period, displacement, velocity, and acceleration of a simple pendulum.
·
To explore the relationship of potential and kinetic energy in a pendulum
·
To determine how the period of a pendulum depends on its maximum displacement.
Physics Laboratory Manual. Jan 1999
Oscillations - Page 1 of 8
Pre-lab Questions
1. The period of a simple pendulum of length L is given by T = 2
pendulum that has a period of 1.57 s?
L/g . What is the length of a
2. The angular displacement of the pendulum bob from equilibrium will be measured by the rotary
motion sensor in radians. If the angle as a function of time is given by θ(t) = (0.50 rad) cos(4.0t),
what is its amplitude, θm, period, T, and frequency, f ?
3. Find the angular velocity of the pendulum in question 2 by taking the derivative of the angle as a
function of time. What is the maximum angular velocity? At what time does this occur?
4. Find the maximum velocity of the pendulum in question 3 assuming it has the length you
calculated in question 1.
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The Human Pendulum
Name______________________________
Partner _____________________________
Equipment
·
·
·
·
Computer with ULI and motion detector
File Swinging
An arm and a leg
Meterstick
Your arm
1. You’ve probably noticed that people swing their arms as they walk. Think about your arm (from your
shoulder joint to finger tips) as a pendulum.
How is your arm similar to the simple pendulum shown on page 1? How is it different?
2. Make sure that the silver cable from the Motion Detector is plugged into Port 2 of the ULI. Start
Logger Pro and open the file Swinging. Have one member of your team swing his or her arm
from the shoulder joint, not bending the arm at the elbow. Let it swing at a natural, comfortable
rate. The other member should hold the motion detector so that it can “see” the motion of the
arm. Don’t forget to keep the detector at least 40 cm from the arm. Swing the arm away from the
body so that the detector sees the arm and not the body.
3. Find the period of the oscillation of the arm. One good method is to find the time required to
make five or ten oscillations and divide by the number of complete cycles.
Period of your arm = _________________ s.
4. The period of a simple pendulum is T = 2 L/g where L is the length of the pendulum from
the pivot to the center of mass. Use this equation and the measurement of the period of your arm
to find the “effective length” of your arm.
Leffective = ________________ m
Where is this length physically located on the arm (i.e., the elbow, forearm, upper arm, etc.)? Does this
make sense? Why?
5. What is the total length of the person’s arm that was used for the measurement? Larm =
_____________ m
How do these two values compare?
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If the arm were a simple pendulum instead of a physical pendulum, would the period be longer or
shorter?
Your leg
1. Your leg, from the hip joint to the foot, can also be considered to be a pendulum. What
difference do you predict there will be between the swinging of an arm and the swinging of a
leg?
2. Have the other member of the team swing his or her leg, moving it from the hip joint, not
bending at the knee. Let it swing at its natural, comfortable rate. The other person should hold
the motion detector so that “sees” the motion of the leg. Find the period of oscillation the same
way that you did for the arm.
Period of your leg = _____________ s
3. How does the period of an arm compare with the measured value for a leg? Explain the
difference.
4. What is the “effective length” of the person’s leg? Leffective = ___________ m
5. Where is this length physically located on the leg (i.e., the knee, lower leg, upper leg, etc.)? Does
this make sense? Why?
6. What is the total length of the person’s leg that was used for the measurement? Lleg =
_____________ m
How do these two values compare?
7. Would you expect a person with shorter or longer legs to walk at a faster pace? Which one
would swing his or her legs with a shorter period?
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A Simple Pendulum
Equipment
·
·
·
·
Computer with ULI and rotary motion sensor
Rod with brass cylinder
File Pendulum
Meterstick and protractor
What does it measure?
1. Disconnect the motion detector from the ULI and connect the black cable from the rotary motion
sensor to Port 2. Open the file Pendulum. With the pendulum hanging straight down, click on
the Zero button. Check the operation of the sensor. While you are collecting data, rotate the rod
to some common angles, 30o, 45o, 90o, 180o (straight up), –90o, etc. Make sure that the sensor
records the correct values in radians.
2. Position the brass cylinder near the end of the rod and tighten the set screw. Pull the rod away
from the equilibrium position. Click on Collect and release the rod. Sketch the graph of angular
position versus time below.
Label the graph with the maximum positive and negative displacements (amplitude) , the time
when the pendulum was at the equilibrium position, and the period.
3. Again displace the rod at about 0.5 rad (30o). Click on Collect and take data. From the angular
position versus time graph find and record the period of the oscillation.
T = ______________ s
4. From the equation for the period of a simple pendulum, calculate the length of a pendulum that
would have this period. Lexpected = ____________ s
5. Measure the length of the pendulum (to the center of the brass cylinder) and record below. Lactual
= _____ m
Do the two values agree? If not, which is larger? What is a possible reason?
6. What is the frequency of oscillation (f = 1/T) f = ___________ oscillations/s
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7. Examine the angular velocity versus time graph. Find the maximum positive and maximum
negative velocities and the minimum velocity of the pendulum.
Magnitude of angular
velocity (rad/s)
ωmin
Angular position(s) of
pendulum (rad)
0
ωmax > 0
ωmax < 0
If the angular position as a function of time is given by the equation θ(t) = θmax cos (2π f t), then
the maximum angular velocity ought to be ωmax = 2π f θmax., where θmax is the amplitude of the
oscillation. Compare the predicted value with the angular velocity found on the graph.
8. Look at the graph of angular acceleration versus time. Find the maximum positive and
maximum negative accelerations and the minimum acceleration of the pendulum.
Magnitude of
angular acceleration
(rad/s2)
αmin
Angular velocity
(rad/s)
Angular position(s) of
pendulum (rad)
0
αmax > 0
αmax < 0
The maximum angular acceleration ought to be αmax = (2π f )2θmax. Compare the predicted value
with the angular acceleration found on the graph.
9. As the pendulum swings back and forth, its energy is changed from one form to another. If
mechanical energy is conserved (work done by friction can be ignored), then the gravitational
potential energy (U = mgh) at the point of maximum angular displacement (where v = 0) is
transformed into kinetic energy (K = ½mv2) when the pendulum is in the equilibrium position (h
= 0).
Find the linear velocity of the brass cylinder when its angular velocity is a maximum (v = Lω).
Measure the mass of the cylinder and calculate the maximum value of its kinetic energy. Kmax =
___________ J.
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10. If energy is conserved, what is the maximum value of the potential energy of the brass cylinder?
Umax = ____________ J
11. Check this value. Measure the vertical displacement (h) of the center of the brass cylinder from
its equilibrium position to its maximum angular displacement.. Does the value of the potential
energy agree with your previous value? Find any percent disagreement of the two values.
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How Small is Small?
Does the period depend on the amplitude of oscillation?
The equation for the period of a pendulum, T = 2 L/g , does not involve either the mass of the
pendulum or its maximum displacement (amplitude). Over what range of amplitudes is this equation
valid? It was derived (see textbook) for displacements that are small enough that sin θ  θ , where θ
is measured in radians) . So, does the period change as the angle is increased? If so, how much? Over what
range of angles can you use this equation?
The experiment
Design an experiment to measure the period of the pendulum as a function of its amplitude. For each initial
angle (from about 0.2 rad to almost 3.14 rad--180o) find the period (measure the time required for 5-10
oscillations). Measure the period for at least six different amplitudes.
Hand in a data table and a graph obtained by using Graphical Analysis for Windows.
Questions
1.
For what angles is the period no more than 10% different from its value at small amplitudes?
2. For what angles is the period within 1% of its value at small amplitudes?
3. The textbook equation for period is based on the assumption that the force on the bob is
proportional to the negative of its displacement from equilibrium. Therefore the acceleration
should also be proportional to the displacement. Using Logger Pro, examine the graph of
angular acceleration versus time for a large-amplitude oscillation. You can change the x-axis
from time to angular position to plot the acceleration versus position. If the acceleration is
proportional to position, then this graph should be a straight line. Is it? For what range of
angles?
4. Explain your results for question 3 using a free-body diagram for the pendulum at small and at
large angles.
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