Simple Harmonic Motion

Oscillations

► Oscillatory motion is a repetitive motion back and forth

about an equilibrium position in a fixed amount of time.

Equilibrium
position

Point B
Point A

4 seconds for
1 complete
cycle

4 seconds for
the 2nd
complete
cycle

Important Oscillatory Motion Terms

► Period (T)– the time it takes to complete one full

cycle.

► The unit for period is seconds.

► In the previous animation, the period was 4 seconds.

► Frequency (f)– the number of cycles per second.

► The unit for frequency is Hertz (1/s)

► In the previous animation, the frequency was 0.25 Hz. This

means it traveled a quarter of the way in one second.

Relationship between Period and
Frequency

► Period and frequency are inverses of each

other.

Example 1

► An FM radio station broadcasts a frequency of 1 x 108 Hz

(or 100 MHz). What is the period?

10 nanoseconds!

Whiteboard Problem

► What is the frequency of an oscillator that has a period

of 0.05 seconds?

f = 20 Hz

Graphing Oscillatory Motion

► If you plot a position versus time graph of an oscillating object

and it creates a sinusoidal graph (sine or cosine function), the
oscillation is describe as simple harmonic motion (SHM).

Sine function

Cosine function

What is Simple Harmonic Motion?

► Simple Harmonic Motion is a simple type of oscillation where

the net force can be described by Hooke’s Law:

► The restoring force is directly proportional to the displacement.

► The restoring force is in the opposite direction of the displacement.

► We will focus on SHM for springs and pendulums.

SHM of Springs

► A spring attached to block m is

stretched with a displacement x and
when released, the spring force
brings the block back towards
equilibrium position.

► The block passes the equilibrium

position because of momentum and
then compresses the spring with a
displacement of x. And spring force
brings it back towards equilibrium.

Spring Amplitude

► The amount the spring is displaced from the equilibrium as it

stretches is the amplitude.

► This is the same displacement (amplitude) as the spring

compression from the equilibrium.

A

A

Energy of the Oscillating Spring

► At the max compression and extension, the mass comes to a temporary

stop, causing the KE to be zero and all energy stored as Us.

► At the equilibrium position (x = 0), the mass is at its greatest speed and

all the energy is KE.

Period of a Spring

Ts – period of spring (s)

m - mass (kg)

k - spring constant (N/m)

Example 3

► A 65kg bungee jumper oscillates up and down on their bungee

cord at the end of their jump. If the bungee cord acts like a
spring with a spring constant of 792 N/m, what will be the
bungee jumper’s period?

Givens:
m = 65kg
k = 792 N/m
Ts = ?

Whiteboard Problem

► A 2kg block is attached to a spring and is undergoing simple

harmonic motion. If the spring constant is 120 N/m, what is the
frequency of the oscillations?

f = 1.23 Hz

Position during SHM

► To find out the position of an object at a given time as it

undergoes simple harmonic motion:

x – position at time t
A – amplitude
f – frequency
t - time

Example 2a

► A 0.9 kg classic wooden toy hangs from a spring that lets it bob

up and down. If the toy is raised up 20cm and released, it
makes 15 complete oscillations in 10 seconds.

a)
What is the period?

Example 2b

► A 0.9kg classic wooden toy hangs from a spring that lets it bob

up and down. If the toy is raised up 20cm and released, it
makes 15 complete oscillations in 10 seconds.

b)
What is the toy’s position at t= 0.80 s?

Example 2c

► A 0.9kg classic wooden toy hangs from a spring that lets it bob

up and down. If the toy is raised up 20cm and released, it makes
15 complete oscillations in 10 seconds.

c)
If the spring constant is 70N/m, what is the maximum speed?

Givens:
m = 0.9kg
x = A = 0.2m
k = 50 N/m

The elastic potential energy at the max stretch is equal to
max kinetic energy at equilibrium.

What is a Pendulum?

► An object that has a

small mass, known as a
pendulum bob, hung
from a pivot point by a
light wire or string.

Restoring Force of a Pendulum

► What is the restoring force of

a pendulum?

► Weight!!!!

► When you lift a pendulum

bob up a certain angle, the
horizontal component of
the weight (mg sinθ) is the
restoring force.

Equilibrium position

θ

mg sinθ

mg

θ

mg cosθ

mg sinθ

SHM of Pendulums

► When the pendulum is displaced

an arc length (s), the restoring
force, weight in x direction, will
bring the pendulum bob back
towards equilibrium. Because the
bob has momentum, it passed
equilibrium and is displaced an
arc length (s) on the other side of
the pendulum.

Pivot point

s

s

SHM of Pendulums

► Important: Pendulums only undergo simple harmonic

motion for small angles (θ). Otherwise the restoring
force is not directly proportional to the displacement (s).

Energy of a Pendulum

Ug = mgy
KE = 0 J

y

y

Ug = mgy
KE = 0 J

Ug = 0 J (at equilibrium)
KE = ½ mv2 (max v)

Period of a Pendulum

Tp – period of pendulum

L – length of string

g – gravitational acceleration

Example 3

► A grandfather clock is designed so that one swing of the

pendulum in either direction takes 1.00 s. What is the length of
the pendulum?

If half the oscillation is 1.00s, then the entire oscillation has a period of 2.00 s.

Givens:
T = 2.00 s
g = 9.8 m/s2

L = ?