Download - Oscillations lecture: AP Mechanics
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8/12/2019 Oscillations lecture: AP Mechanics
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Given a potential energy graph, oscillations will occur between turning points
determined by
txUE
E
tx
Even the most asymmetric well can be approximated by a parabola if the systemstays close to the bottom.
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8/12/2019 Oscillations lecture: AP Mechanics
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2
2
1kxU
Simple Harmonic Motion k and x need not refer tospring parameters, but the
spring-mass system is an
important example
kx
dx
dUF
2
2
dt
xdmmakx
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02
2
xm
k
dt
xdStandard form of harmonic oscillator equation
Solution: tAtx cos mk2
andA determined by initial conditions
Exercise: A spring-mass with k = 25 N/m and m = 5 kg is pulled out 2 m and released.
Find its position at any later time.
amplitude phase angle
phase
angular frequency
2cos0 Ax 0sin0 Av
tAtv sin
2
0
A
ttx 5cos2
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The motion will have completed one full cycle when the phase has
increased by 2. This time interval is called the periodof the oscillation.
2
tTt
2T
Energy Considerations
How can the total energy be constant when both K and U depend on time?
22
2
1
2
1kxmvE
22 cos2
1sin2
1 tAktAmE
ttkAE 222 cossin2
1 2
2
1kAE
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Vertical Springs
When a spring mass-system
vibrates vertically, an extra
force, gravity is involved.How does this affect the
motion?
k
mx
m
kx
kxmgnetF
gxm
k
dt
xd
2
2
? Rewrite: 02
2
k
mgx
m
k
dt
xd
Define new displacement:kmgxx Note that:
2
2
2
2
dtxd
dtxd
Then we have: 02
2
xm
k
dt
xdGravity does not change the period.
All effects of gravity can be ignored if we choose to measure X from the newequilibrium point.
mg
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Example: A 1 kg mass hangs vertically from a spring with k = 20 N/m. A .1 kg dart
moving at 20 m/s is fired into the mass and sticks into it. Find the period and
amplitude of the resulting harmonic motion.
kg1
fPP 0
V1.1201.
smV 82.1
The new equilibrium point is
mx
f55.
20
101.1
0
The period just depends on k and the mass oscillating:
s
k
mT 47.1
20
1.122
Use energy conservation with x referred to new equilibrium point:
fEE 0
222 202
105.20
2
182.11.1
2
1A
mA 43.
The old equilibrium point is
mxi 50.
20
1010
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Other Oscillators
Torsion Pendulum
As wire twists, a restoring
torque is exerted.
k
I
Fixed axis rotation
I
2
2
dt
dIk
0
2
2
I
k
dt
d
k
IT 2 tt mcos
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Simple Pendulum
T
mg
L
ILmg sin
2mLI
0sin2
2
L
g
dt
d? Small angle approximation:
sin
02
2
L
g
dt
d
g
LT 2
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Compound Pendulum
H
CM IHmg sin
0sin2
2
I
mgH
dt
d
Small angle approximation:
sin0
2
2
I
mgH
dt
d
mgH
IT 2
About pivot