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Simple Harmonic MotionSimple Harmonic Motion Vibrations
Vocal cords when singing/speakingString/rubber band
Simple Harmonic MotionRestoring force proportional to displacementSprings F = -kx
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Question IIQuestion IIA mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest?
1. When x = +A or -A (i.e. maximum displacement)
2. When x = 0 (i.e. zero displacement)
3. The acceleration of the mass is constant
+A
t-A
x
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Potential Energy in SpringPotential Energy in Spring Force of spring is Conservative
F = -k xW = -1/2 k x2
Work done only depends on initial and final position
Define Potential Energy PEspring = ½ k x2
Force
x
work
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***Energy in SHM******Energy in SHM*** A mass is attached to a spring and set to
motion. The maximum displacement is x=AEnergy = PE + KE = constant!
= ½ k x2 + ½ m v2
At maximum displacement x=A, v = 0Energy = ½ k A2 + 0
At zero displacement x = 0
Energy = 0 + ½ mvm2
Since Total Energy is same
½ k A2 = ½ m vm2
vm = sqrt(k/m) Am
xx=0
0x
PES
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Question 3Question 3A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest?
1. When x = +A or -A (i.e. maximum displacement)
2. When x = 0 (i.e. zero displacement)
3. The speed of the mass is constant
+A
t-A
x
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Question 4Question 4 A spring oscillates back and forth on a frictionless horizontal
surface. A camera takes pictures of the position every 1/10th of a second. Which plot best shows the positions of the mass.
EndPoint EndPointEquilibrium
EndPoint EndPointEquilibrium
EndPoint EndPointEquilibrium38
1
2
3
X=0
X=AX=-A
X=A; v=0; a=-amax
X=0; v=-vmax; a=0
X=-A; v=0; a=amax
X=0; v=vmax; a=0
X=A; v=0; a=-amax
Springs and Simple Harmonic Springs and Simple Harmonic MotionMotion
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What does moving in a circle have to do with moving back & forth in a straight line ??
y
x
-R
R
0
1 1
2 2
3 3
4 4
5 5
6 62
R
8
7
8
7
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x
x = R cos = R cos (t)since = t
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Simple Harmonic Motion:Simple Harmonic Motion:
x(t) = [A]cos(t)
v(t) = -[A]sin(t)
a(t) = -[A2]cos(t)
x(t) = [A]sin(t)
v(t) = [A]cos(t)
a(t) = -[A2]sin(t)
xmax = A
vmax = A
amax = A2
Period = T (seconds per cycle)
Frequency = f = 1/T (cycles per second)
Angular frequency = = 2f = 2/T
For spring: 2 = k/m
OR
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ExampleExample
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
Which equation describes the position as a function of time x(t) =
A) 5 sin(t) B) 5 cos(t) C) 24 sin(t)
D) 24 cos(t) E) -24 cos(t)
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ExampleExample
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
What is the total energy of the block spring system?
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ExampleExample
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
What is the maximum speed of the block?
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ExampleExample
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
How long does it take for the block to return to x=+5cm?
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Pendulum MotionPendulum Motion For small angles
T = mgTx = -mg (x/L) Note: F proportional to x!
Fx = m ax
-mg (x/L) = m ax
ax = -(g/L) xRecall for SHO a = -2 x
= sqrt(g/L) T = 2 sqrt(L/g)
Period does not depend on A, or m!
m
L
x
T
mg
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Example: ClockExample: Clock If we want to make a grandfather clock so that the
pendulum makes one complete cycle each sec, how long should the pendulum be?
Question 1Question 1Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should:
1. Make the pendulum shorter
2. Make the pendulum longer
Lg
gL
T 2
2
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