Download simple harmonic motion – the pendulum and the spiral spring

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EP 236 WAVE LABORATORY
1. SIMPLE HARMONIC MOTION and COUPLED OSCILLATIONS
PURPOSE
This experiment consist of from two two part. In the first part, we examine the simple harmonic
motion examples of the simple pendulum and the spiral spring. Subsequently, the superposition of
the fundamental oscillations of coupled pendulums will observed in three steps:
Coupling two similar harmonic oscillators (rod pendulums) with a natural frequency f 0 by
means of bifilar pendulum.
Realization of the first and second fundamental oscillations by deflecting the pendulums in
the same and opposing directions; determination of the respective frequencies f 1 and f2.
Superposition of fundamental oscillations to form beats; determining the frequency f 3 of each
individual pendulum and the beat frequency fb.
APPARATUS
Different metal spheres, both of the same diameter, strong thread, long support rod and clamp,
meter stick, stop clock, the spiral springs, weight holder and slotted weights. 1 Double pendulum, 1
stop-clock, 1 metal scale (1 m), 1 steel ball with eye, 1 set of 12 marking pins, approximately 90
cm fishing line, 1 clamping plug, 2 stand rods (1.5 m), 1 stand rod (1m), 2 stands rod (25 cm), 1
insulated stand rod, 7 leybold multiclamps, 2 bench clamps, 1 small stand base.
A. Simple Pendulum and The Spiral Spring
Any motion which repeats itself in equal intervals of time is called periodic motion. The simplest
kind of periodic motion is perhaps uniform circular motion. Other common examples of simple
periodic motion are: The pendulum of clock, the to- and –fro motion of childs swing, and the
vibration of a tunning fork.
Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither
driven nor damped. The motion is periodic, as it repeats itself at standard intervals in a specific
manner - described as being sinusoidal, with constant amplitude. It is characterised by its amplitude
which is always positive and depends on how motion starts initially, its period which is the time for a
single oscillation and its phase which depends on displacement as well as velocity of the moving
object.
The position of a simple harmonic motion oscilator varies periodically in time according to the
expression
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(1.1)
where x is the displacement, A is the amplitude of oscillation, ω is the angular velocity, t is the
elapsed time, and φ is the phase of oscillation. If there is no displacement at time t = 0, the phase
φ = 0.
The time T for one complete vibration is called the period of the motion
(1.2)
PART 1. THE PENDULUM
A mass M suspended by a light string of length L from a fixed point. It provides an elegant example
of simple harmonic motion. The oscillations of a pendulum are governed by the following equation:
(1.3)
For small angles of oscillations sin =x/l, is governed by:
(1.4)
The frequency (f) and the period (T) of the pendulum in this case are given by:
(1.5)
Therefore, the simple pendulum has the following properties. i) The period of a pendulum is
independent of its own mass and the amplitude of the oscillation. ii) The period of the pendulum is
depends on its length.
Figure 1. The Simple Pendulum
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Engineering of Physics Department
PART 2: THE SPRING
The simple harmonic motion that occurs when a spring is combined with a known mass, m. The
most common object that obeys Hook’s law on a large scale is a spring. If the net force on the
particle in a periodic motion is such that the magnitude of the force is proportional to the
displacement of the particle but the direction of the force is always opposite to that of the
displacement (the force is always directed toward the midpoint). This is called Hook’s law. Namely,
(1.6)
where k is spring constant, x is the displacement.
The period T of the motion depends on the spring constant k and the mass m in the following
equation,
(1.7)
The schematic diagram of the spring-mass system is shown in figure 2. Here the motion of the mass
is up and down on astraight line.
Figure 2. The Spring
The spring combinations can be formed by adding two or more springs together in series or parallel.
According to the type of connections, the spring constants can be derived.
PROCEDURE AND CALCULATIONS
PART 1. The Simple Pendulum
The experiment consists of determining the frequency (period) of various pendulums of different M
and L, for several initial angles of displacement. The equipment includes pendulum bobs of two
quite different masses (steel and aluminum) and a string whose length you can vary.
1. Weigh the two masses.
2. Adjust the string length.
3. Initiate oscillations by displacing the mass with a small angle
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4. Determine the period by timing oscillations. Each oscillation involves a complete swing back
to the starting position. The period T is determined by counting the number N of complete
oscillations in a time t:
5.
6.
7.
8.
PART
1.
2.
3.
4.
Calculate the period by using the length of the string.
Compare these two values of the period and calculate percentage difference.
Calculate the acceleration of gravity by using the T value found in 4 and the string length.
Plot the graph T2 vesus L
2. The Spring
Suspend the spring in vertical position and measure the position of the spring.
Place a mass M at the end of the spring. Record the equilibrium position of the mass.
Calculate the spring constant, k.
With the mass, M, still attached to the spring, pull the mass straight up a distance and record
this distance and release it. You should see the mass oscillate up and down in SHM.
5. Use a stopwatch to measure the time it takes to complete 10 full cycles of SHM.
6. Calculate the period, T, of the oscillator.
7. Calculate the angular velocity, ω, of the SHM by using the known mass and the spring
constant obtained in 3.
8. Calculate the angular velocity, ω, of the SHM again by using the period T calculated in 6.
9. Compare these two values of angular velocities and calculate percentage difference.
10. What is the amplitude?
11. Calculate the maximum speed of the SHM.
12. Calculate the maximum acceleration of the SHM.
B. The Coupled Oscillations
If one pendulum is started swinging with small amplitude, the other slowly builds up amplitude as
the spring feeds energy from the first into the second. Then the energy flows back into the first and
the cycle repeats. The general behavior can be quite complicated, depending on the initial excitation
as well as the system parameters, but a particularly simple situation can be set up for two identical
pendula. If you start the two swinging together they will continue to swing in unison at their natural
frequency. Alternatively, if they are started exactly out of phase (swinging in opposite directions),
they will maintain this motion, but at a higher frequency than they would oscillate if uncoupled.
These two possibilities are called the normal modes of the system, and are illustrated in Figure 3.
When the pendula are not identical there are still two normal modes, but the motions are more
complicated and neither mode is at the uncoupled frequency.If we observe two pendulums where a
coupling spring is acting on the pendulum masses m1=m2=m with the sprig constant k, the spring’s
force F1 and F2 acts on each mass as shown in Fig. 1 when the system is deflected from its rest
position in addition to the weight component
F1
k 0 x1
k ( x1
x2 )
F2
k 0 x2
k ( x2
x1 )
where
k0
mg
l
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Engineering of Physics Department
Fig. 3 Normal modes of coupled pendululum
It therefore follows in the special case of in-phase oscillations (x1=x2) or oscillations with phase
displacement of 1800 (x1=-x2) that
F1
F2
F1
k 0 x1
(k 0
F2
(k 0
where x1
2k ) x1
x2
where x2
2k ) x1
x1
(1.8)
(1.9)
In addition to the linear law of forces stated in (1.8) and (1.9), Newton’s law of motion applies to
the pendulum movements:
F
xw2
where a
ma
(1.10)
The angular velocities are given by (1.8) or (1.9) and (1.10):
k0
m
w1
(1.11)
w0
in the case of in-phase oscillations (1st fundamental oscillation) and
k0
w2
2k
m
w12
2k
m
(1.12)
in the case of opposite-phase oscillations (2nd fundamental oscillation).
This implies that the angular velocity w1 is grater than w2, and thus that the frequency f2 of the 2nd
fundamental oscillation is greater than the frequency f1 of the 1st fundamental oscillation occurring
at the natural frequency of f0.
The movement of the coupled pendulums is described by a system of simultaneous
differential equations
d 2 x1
m
dt 2
d 2 x2
m
dt 2
or where
k0
k 0 x1
k ( x1
k 0 x2
k ( x2
x2 )
x1 )
w12 m
d 2 x1
dt 2
d 2 x2
dt 2
k
( x1 x 2 )
m
k
( x 2 x1 )
m
With the initial conditions for a maximum deflection xm of a pendulum with stationary second
pendulum at a time t=0,
dx1
dt
dx2
dt
The equation produces the solutions
0;
x1
xm ; x2
0;
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xm
(cos(w1t ) cos(w2 t ))
2
xm
(cos(w1t ) cos(w2 t ))
2
x1
x2
Using the expressions for the sum of differences of two angular functions,
a b
a b
cos
2
2
a b
a b
2 sin
sin
2
2
cos a cos b
2 cos
cos a cos b
We obtain the distance-time equations for the movement of coupled pendulums for the abovementioned initial conditions:
x1
x2
x m cos(
x m sin(
w2
w1
2
w2
w1
2
t ) cos(
t ) sin(
w2
w1
2
w2
w1
2
t)
(1.13)
t)
The equations (1.13) describe oscillations produced by the superposition of fundamental oscillations
with angular frequencies of w 1 and w2.
A harmonic oscillation is produced with an angular frequency given by
w1
w3
w2
2
Or the frequency
f3
f1
f2
(1.14)
2
The amplitude fluctuates between 0 and the maximum value xm with an angular frequency
wb
w2
w1
Or the frequency
fb
f2
f1
(1.15)
Both pendulums carry out beats which are displaced in respect of each other by T b/2. In accordance
with (1.15), the following relationship exists between the oscillation periods of the fundamental
oscillations T1, T2 and the beat period Tb;
Tb
Each pendulum carries out a phase jump
1
f2
f1
T1 .T2
T1 T2
(1.16)
, after having reached zero amplitude and having started
to oscillate again.
PROCEDURE:
Following the below steps to carry out this experiment.
1. Suspend the pendulums for marking pins as shown in Figure 4. and carefully secure in
clamps.
2. Push both pendulum masses down right to the bottom so that their bases are exactly at the
ends of the pendulum rods.
3. Secure the hook (a) for the coupling pendulum to both pendulums at exactly the same
height (approximately 15 cm above the pendulum masses).
4. After positioning the hooks (a), but before coupling, check that the natural oscillations of the
pendulums match, and balance exactly if necessary by slightly moving one pendulum mass
(both pendulums should continue to oscillate synchronously after being deflected uniformly).
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Engineering of Physics Department
5. Guide the fishing line of the comparison pendulum through the perpendicular bore (b) with
the plug pressed in; adjust the pendulum length so that the oscillation period is equal to that
of the uncoupled pendulum.
xm
xm
xm xm
Figure 4. Schematic diagram of coupled oscillation experiment
CALCULATIONS
i) Natural Oscillation (without coupling)
1. Measure the time 10 T0 for 10 oscillations of a pendulum several times to determine the
natural frequency f0.
2. Calculate f0=1/T0 from the mean value of the values measured for T 0.
ii) Fundamental Oscillations
1. Attach the coupling device in accordance with Figure 4; deflect the pendulums in the same
direction and at the same amplitude (amplitude approximately 6 cm), as shown in inset
diagram A, and allow it to oscillate.
2. Measure the time 10 T1 for 10 oscillations several times.
3. Calculate the frequency f1 of the first fundamental oscillation from the mean value for T 1; in
the same way, determine the frequency f 2 for the 2nd fundamental oscillation, at which the
pendulums oscillate in antiphase with the same amplitude (see inset diagram B).
iii) Beats
1. Deflect pendulum 1 while holding pendulum 2 stationary (see inset diagram C), and release
both pendulums simultaneously; firstly measure the time 5T3 for 5 oscillations of 1 pendulum
several times and calculate the frequency f3 from the mean value.
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Engineering of Physics Department
2. Then measure the time Tb taken for pendulum 2, suspended in equilibrium at the start of
oscillation, to stop for the first time after one beat has passed through; measure Tb at least 5
times and determine the beat frequency fb from the mean value.
3. Allow comparison pendulum 3 to oscillate in order to demonstrate the phase jump; while
holding pendulum 2 stationary, deflect pendulum 1 to the left and then release both
pendulums exactly when the comparison pendulum reaches the left stationary point (see
inset diagram C), so that pendulums 1 and 3 oscillate in phase at first; compare the phase
position of pendulum 1 with that of pendulum 3 when it is stationary for the second time.
Note: The comparison pendulum is matched to the natural frequency of the uncoupled pendulum
(see “procedure”), so that its oscillation period does not exactly correspond with that of the coupled
pendulums 1 and 2. However, this does not impair demonstration of the same jump. Furthermore,
w0 and w3 only displace a small amount in the case of weak coupling (k << k0).
DISCUSSION and CONCLUSION
1) What is the magnitude of the acceleration of an oscillator of amplitude A and frequency f
when its speed is maximum? When its displacement is maximum?
2) Is a bouncing ball an example of simple harmonic motion? Is the daily movement of a
student from home to school and back simple harmonic motion? Why or why not?
3) Does the displacement of an oscillating particle between t = 0 and a later time t necessarily
equal the position of the particle at time t? Explain.
4) Determine whether or not the following quantities can be in the same direction for a simple
harmonic oscillator: (a) position and velocity, (b) velocity and acceleration, (c) position and
acceleration.
5) A block–spring system undergoes simple harmonic motion with amplitude A. Does the total
energy change if the mass is doubled but the amplitude is not changed? Do the kinetic and
potential energies depend on the mass? Explain.
6) What happens to the period of a simple pendulum if the pendulum’s length is doubled? What
happens to the period if the mass of the suspended bob is doubled?
7) A simple pendulum is suspended from the ceiling of a stationary elevator, and the period is
determined. Describe the changes, if any, in the period when the elevator (a) accelerates
upward, (b) accelerates downward, and (c) moves with constant velocity.
8) Imagine that a pendulum is hanging from the ceiling of a car. As the car coasts freely down a
hill, is the equilibrium position of the pendulum vertical? Does the period of oscillation differ
from that in a stationary car?
9) A simple pendulum undergoes simple harmonic motion when θ is small. Is the motion
periodic when θ is large? How does the period of motion change as θ increases?
10)If a grandfather clock were running slow, how could we adjust the length of the pendulum to
correct the time?
11) Write a techological examples of coupled oscillators.
12)If the both pendula vibrate inphase with the same amplitude and with the same frequency,
what can you say about the angular characteristic frequency?.
13) If the both pendula vibrate “in opposite phase” with the same amplitude and with the same
frequency, what can you say about the angular characteristic frequency?
14)The length of the string or wire supporting a pendulum increases slightly when its
temperature is raised. How would this affect a clock operated by a simple pendulum?