Simple Harmonic Motion
The Ideal Spring and Simple Harmonic Motion
FxApplied  k x
spring constant
Units: N/m
Sample Problem
Hooke’s Law
If a mass of 0.55 kg attached to a vertical
spring stretches the spring 2.0 cm from its
original equilibrium position, what is the
spring constant?
mg
(0.55 kg)(9.81 m/s2 )
k

x
–0.020 m
k  270 N/m
Simple Harmonic Motion
Simple harmonic motion describes any
periodic motion that is the result of a restoring
force that is proportional to displacement.


The motion of a vibrating mass-spring system is an
example of simple harmonic motion.
Because simple harmonic motion involves a
restoring force, every simple harmonic
motion is a back-and-forth motion over the
same path.
The Simple Pendulum

A simple pendulum consists of
a mass called a bob, which is
attached to a fixed string.
The forces acting on the bob
at any point are the force
exerted by the
string and the
gravitational force.
Amplitude, Period, and Frequency

Amplitude of the vibration is the
maximum displacement from equilibrium.

A pendulum’s


For a mass-spring system,


measured by the angle between the pendulum’s
equilibrium position and its maximum
displacement.
the maximum amount the spring is stretched or
compressed from its equilibrium position.
SI units are radian (rad) and meter (m).

The period (T) is the time that it takes a
complete cycle to occur.


The SI unit of period is seconds (s).
The frequency (f) is the number of cycles or
vibrations per unit of time.


The SI unit of frequency is hertz (Hz).
Hz = s–1
1
1
f  or T 
T
f
Period of a Simple Pendulum

The period of a simple pendulum
depends on the length and on the freefall acceleration.
L
T  2
ag

length
period  2
free-fall acceleration
The period does not depend on the
mass of the bob or on the amplitude (for
small angles).
Period of a Mass-Spring System

The period of an ideal mass-spring
system depends on the mass and on
the spring constant.
m
T  2
k
mass
period  2
spring constant
The period does not depend on the
amplitude.
 This equation applies only for systems in
which the spring obeys Hooke’s law.

Simple Harmonic Motion and the
Reference Circle
DISPLACEMENT
x  A cos  A cos t
Amplitude
Constant angular speed
(rad/s)
Simple Harmonic Motion and the Reference Circle
x  A cos  A cos t
Amplitude
Constant angular speed
(rad/s)
amplitude A: the maximum displacement
period T: the time required to complete one cycle
frequency f: the number of cycles per second (measured in Hz)
1
f 
T
2
  2 f 
T
The Maximum Speed of a Loudspeaker
Diaphragm
The frequency of motion is 1.0 KHz and the amplitude is
0.20 mm.
(a)What is the maximum speed of the diaphragm?
(b)Where in the motion does this maximum speed
occur?
v x  vT sin    
A sin t
vmax
(a)



vmax  A  A2 f   0.20 10 3 m 2  1.0 103 Hz

 1.3 m s
(b) The maximum speed
occurs midway between
the ends of its motion.