Transcript Document

Oscillatory Motion
Serway (Chap.15)
Oscillatory Motion
Motion in the real world may not fit some of our earlier models
(linear or circular motion, uniform acceleration).
Many phenomena are repetitive or oscillatory.
Example: Block and spring
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Equilibrium: no net force
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The spring force is always directed back
towards equilibrium. This leads to an
oscillation of the block about the
equilibrium position.
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F = -kx
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x
For an ideal spring, the force is
proportional to displacement. For this
particular force behaviour, the oscillation
is simple harmonic motion.
Simple Harmonic Motion
x(t)
t
In Simple Harmonic Motion (SHM), the displacement is a
sinusoidal function of time, e.g.,:
x(t )  A cos t
or
x(t )  A sin t
Questions: Is a bouncing ball described by SHM, even if it
returns to the same height?
Is it periodic motion?
x(t)
t
In general,
x(t )  A cos(t  f )
Phase
The quantity (t + f) is called the phase, and is measured in radians.
The cosine function traces out one complete cycle when the phase
changes by 2p radians. The phase is not a physical angle!
Three constants specify the motion are:
1) Amplitude, A
2) Angular Frequency, 
3) Initial phase (or phase constant), f
SHM:
x(t)
A
T
x  A cos(t  f )
t
-A
A is the maximum value of x (x ranges from +A to -A).
f
gives the initial position at t=0: x(0) = A cosf .
 is related to the period T and the frequency f = 1/T :
T (period) is the time for one complete cycle (seconds).
Frequency f (cycles per second or hertz, Hz) is the number
of complete cycles per unit time.
The period T of the motion is the time needed to complete one cycle:
and
so
x(t) = A cos (t + f)
cos (f) = cos(2p + f)
x (T) = x(0) if T = 2p radians (or 360°)
2p

 2pf
T
Units: radians/second or s-1
Example 1
The block is displaced a distance x = 5 cm
from its equilibrium position and released from
rest at time t = 0. Its motion is SHM with
period 2 seconds. Write the function x(t).
Steps:
1) Sketch a graph.
2) Write x = A cos (t  f).
3) Evaluate A, , and f.
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x
Example 2
The block is at its equilibrium position and
is set in motion by hitting it (and giving it
an initial velocity) at time t = 0. Its motion
is SHM with amplitude 5 cm and period 2
seconds. Write the function x(t).
v0
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x
QUIZ
At t=0, a block is at x0 = +5 cm, with
positive velocity v0. Its motion is SHM with
amplitude 10 cm and period 2 seconds. If
x(t) = A cos (t  f), the phase constant f
should be:
A)
B)
C)
D)
E)
0o
30o
60o
-30o
-60o
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x0
v0
QUIZ
A block on the end of a spring is pulled to
position x=A and released from rest. In
one full cycle of its motion, through what
total distance does it travel?
A) ½ A
B) A
C) 2A
D) 4A
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A