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Simple Harmonic Motion
Simple Harmonic Motion
 Vibrations
Vocal cords when singing/speaking
String/rubber band
 Simple
Harmonic Motion
Restoring force proportional to displacement
Springs F = -kx
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Question II
A 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
x
+A
t
-A
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Potential Energy in Spring
 Force
of spring is Conservative
F = -k x
W = -1/2 k x2
Force
work
x
Work done only depends on initial and
final position
Define Potential Energy PEspring = ½ k x2
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***Energy in SHM***
A
mass is attached to a spring and set to
motion. The maximum displacement is x=A
Energy = PE + KE = constant!
= ½ k x2 + ½ m v2
At maximum displacement x=A, v = 0
PE
2
Energy = ½ k A + 0
S
At zero displacement x = 0
Energy = 0 + ½ mvm2
Since Total Energy is same
0
2
2
½ k A = ½ m vm
m
vm = sqrt(k/m) A
x=0
x
x
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Question 3
A 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
x
+A
t
-A
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Question 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.
1
EndPoint
Equilibrium
EndPoint
EndPoint
Equilibrium
EndPoint
EndPoint
Equilibrium
EndPoint
2
3
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Springs and Simple Harmonic
Motion
X=0
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
X=-A
X=A
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What does moving in a circle have to do with
moving back & forth in a straight line ??
x = R cos  = R cos (wt)
since  = w t
x
x
1
1
2
R
3
R
8

2
8
7
3
y
7
4
6
5
0
-R

2

4

3
2
6
5
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SHM and Circles
Simple Harmonic Motion:
x(t) = [A]cos(wt)
v(t) = -[Aw]sin(wt)
a(t) = -[Aw2]cos(wt)
x(t) = [A]sin(wt)
OR
v(t) = [Aw]cos(wt)
a(t) = -[Aw2]sin(wt)
xmax = A
Period = T (seconds per cycle)
vmax = Aw
Frequency = f = 1/T (cycles per second)
amax = Aw2
Angular frequency = w = 2f = 2/T
For spring: w2 = k/m
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Example
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(wt)
B) 5 cos(wt)
C) 24 sin(wt)
D) 24 cos(wt)
E) -24 cos(wt)
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Example
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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Example
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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Example
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 Motion
 For
small angles
T = mg
Tx = -mg (x/L)
Note: F proportional to x!
S Fx = m ax
-mg (x/L) = m ax
ax = -(g/L) x
Recall for SHO a = -w2 x
w = sqrt(g/L)
T = 2  sqrt(L/g)
Period does not depend on A, or m!
L
T
x
m
mg
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Example: 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 1
Suppose 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
g
w
L
T
2
L
 2
w
g
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Summary
 Simple
Harmonic Motion
Occurs when have linear restoring force F= -kx
 x(t) = [A] cos(wt)
v(t) = -[Aw] sin(wt)
a(t) = -[Aw2] cos(wt)
 Springs
F = -kx
U = ½ k x2
w = sqrt(k/m)
 Pendulum
(Small oscillations)
w = sqrt(L/g)
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