Oscillatory Motion

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Oscillatory Motion
Serway Jewett (Chapter 15)
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Equilibrium position no net force
M
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SHM
x(t)
T
A
A amplitude f phase constant w angular
frequency
t
-A
A is the maximum value of x (x ranges from A to
-A). f gives the initial position at t0
x(0) A cosf . w 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.
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units radians/second or s-1
? (omega) is called the angular frequency of
the oscillation.
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Velocity and Acceleration
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Position, Velocity and Acceleration
x(t) t
v(t) t
a(t) t
Question Where in the motion is the velocity
largest? Where in the motion is acceleration
largest?
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Example
SHM can produce very large accelerations if the
frequency is high. Engine piston at 4000 rpm,
amplitude 5 cm
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Simple Harmonic Motion
SHM
We can differentiate x(t)
and find that acceleration is proportional to
displacement
a(t) - w 2 x(t)
But, how do we know something will obey
xAcos(?t) ???
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Mass and Spring
F -kx
Newtons 2nd Law
M
so
x
This is a 2nd order differential equation for the
function x(t). Recall that for SHM, a -w 2 x
identical except for the proportionality
constant. So the motion of the mass will be SHM
x(t) A cos (wt
f), and to make the equations for acceleration
match, we require that
, or
(and w 2p f, etc.).
Note The frequency is independent of amplitude
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Example Elastic bands and a mass. A mass, m, is
attached to two elastic bands. Each has tension
T. The system is on a frictionless horizontal
surface. Will this behave like a SHO?
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Quiz
The ball oscillates vertically on a single spring
with period T0 . If two identical springs are
used, the new period will be

1. longer
2. shorter

by a factor of

2. 2
3. 4

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Quiz
The ball oscillates vertically on a single spring
with period T0 . If two identical springs are
used, the period will be

1. longer
2. shorter

by a factor of

2. 2
3. 4
4. 1

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Quiz
µs0.5
B
?10 s-1

• The amplitude of the oscillation gradually
  increases till block B starts to slip. At what A
  does this happen?(there is no friction between
  the large block and the surface)
• Any A
• 5 cm
• 50 cm
• Not known without mB.

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Solution
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Energy in SHM
Look again at the block spring
We could also write E KU ½ m(vmax )2
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E
U, K oscillate back and forth out of phase with
each other the total E is constant. n.b.! U, K
go through two oscillations while the position
x(t) goes through one.
K
U
t
T
x
t
v
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Suppose you double the amplitude of the motion

• 1) What happens to the maximum speed?
• Doubles
• 4 x Larger
• Doesnt change

• 2) What happens to the maximum acceleration?
• Doubles
• 4 x Larger
• Doesnt change

• 3) What happens to the the total energy?
• Doubles
• 4 x Larger
• Doesnt change

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Summary
SHM
(get v, a with calculus)
Definitions amplitude, period, frequency,
angular frequency, phase, phase constant.
The acceleration is proportional to
displacement a(t) -w2
x(t)