Periodic Motion
Simple periodic motion is that motion in which a
body moves back and forth over a fixed path,
returning to each position and velocity after a
definite interval of time.
1
f 
T
Amplitude
A
Period, T, is the time
for one complete
oscillation. (seconds,s)
Frequency, f, is the
number of complete
oscillations per
second. Hertz (s-1)
Example 1: The suspended mass makes 30
complete oscillations in 15 s. What is the
period and frequency of the motion?
15 s
T
 0.50 s
30 cylces
x
F
Period: T = 0.500 s
1
1
f  
T 0.500 s
Frequency: f = 2.00 Hz
Simple Harmonic Motion, SHM
Simple harmonic motion is periodic motion in
the absence of friction and produced by a
restoring force that is directly proportional to
the displacement and oppositely directed.
x
F
A restoring force, F, acts in
the direction opposite the
displacement of the
oscillating body.
F = -kx
Hooke’s Law
When a spring is stretched, there is a restoring
force that is proportional to the displacement.
F = -kx
x
m
F
The spring constant k is a
property of the spring given by:
k=
DF
Dx
Displacement in SHM
x
m
x = -A
x=0
x = +A
• Displacement is positive when the position is
to the right of the equilibrium position (x = 0)
and negative when located to the left.
• The maximum displacement is called the
amplitude A.
Period and Frequency as a Function
of Mass and Spring Constant.
For a vibrating body with an elastic restoring force:
Recall that F = ma = -kx:
1
f 
2
k
m
m
T  2
k
The frequency f and the period T can be found if
the spring constant k and mass m of the vibrating
body are known. Use consistent SI units.
The Simple Pendulum
The period of a simple
pendulum is given by:
L
T  2
g
L
For small angles q.
1
f 
2
g
L
mg
Summary
Simple harmonic motion (SHM) is that motion in
which a body moves back and forth over a fixed
path, returning to each position and velocity
after a definite interval of time.
The frequency (rev/s) is the
reciprocal of the period (time
for one revolution).
x
m
F
1
f 
T
Summary (Cont.)
Hooke’s Law: In a spring, there is a restoring
force that is proportional to the displacement.
F  kx
x
The spring constant k is defined by:
m
F
DF
k
Dx
Summary (SHM)
x
a
v
m
x = -A
x=0
F  ma  kx
x = +A
 kx
a
m
Conservation of Energy:
½mvA2 + ½kxA 2 = ½mvB2 + ½kxB 2
Summary (SHM)
1
2
mv  kx  kA
2
1
2
k
v
A2  x 2
m
x  A cos(2 ft )
2
1
2
v0 
2
k
A
m
a  4 f x
2
v  2 fA sin(2 ft )
2
Summary: Period and
Frequency for Vibrating
Spring.
a
v
x
m
x = -A
x=0
x = +A
1
f 
2
a
x
x
T  2
a
1
f 
2
k
m
m
T  2
k
Summary: Simple Pendulum
and Torsion Pendulum
1
f 
2
g
L
I
T  2
k'
L
L
T  2
g