Simple Harmonic Motion for Grade 12

1. Whenever the force acting on an object is:
1. Proportional to the displacement
2. In the opposite direction,
the object exhibits simple harmonic motion
(SHM).
Examples
1. mass attached to a spring
2. simple pendulum.

2. Definitions of Terms
• Amplitude = A = the maximum displacement of
the moving object from its equilibrium position.
• Period = T = the time it takes the object to
complete one full cycle of motion.
• Frequency = f = the number of cycles or
vibrations per unit of time.

3. Vertical Spring
Mass Attached to a Spring
m x = 0 “Equilibrium Position”
x > 0
x < 0
x = displacement from
equilibrium

4. Period of an object on a vertical spring
exhibiting SHM is:
2
m
T
k


T = period
m = mass of object
K = spring constant

5. Force always opposite the displacement from equilibrium
 If we stretch a spring
with a mass on the end
and let it go, the mass
will oscillate back and
forth (if there is no
friction).
 This oscillation is called
Simple Harmonic
Motion, because F is a
restoring force.
Horizontal
Spring
F kx
 

6. As previously stated, a simple harmonic oscillator is any object
that oscillates and is subject to a restoring force. Example:
horizontal mass on the end of a spring. F is a linear restoring
force. Hooke’s law F= -kx applies
F
X
F kx
 

7. The frequency and period of the simple harmonic
oscillator are independent of the amplitude.
2
m
T
k



8. t
Sine or cosine curve representation of a restoring
force and simple harmonic motion.
2
m
T
k



9. Another View
2
m
T
k



11. Case 2 - The Simple Pendulum. A component of
the weight acts as the restoring force
Component of
weight restoring
mass to
equilibrium
mg sin

12. Period for The Simple Pendulum:
A pendulum is made by suspending a mass m at the end of a string
of length L. The period of oscillation for small displacements is
given by the following formula.
2
l
T
g


T = period
L = length
“g”= acceleration due to gravity

14. Period of a simple harmonic oscillator
representation in the form of a cosine curve.
T/4 = time for
quarter cycle
T/2 = time for
half cycle
3T/2 = time for three
quarters of a cycle
2
l
T
g



15. Simple Harmonic Motion and Circular Motion, compared
to circular motion
2
l
T
g



16. Energy in Simple Harmonic Motion