Simple Harmonic Motion

Definitions

• Periodic Motion – When a vibration or

oscillation repeats itself over the same

path

• Simple Harmonic Motion – A specific form

of periodic motion in which the restoring

force is proportional to distance from the

equilibrium position.

Objects that exhibit SHM

• Spring Systems

• Pendulums

• Circular Motion

• Waves

– Sound, Light, Pressure

Definitions

• Period – Time required for one complete

cycle (Seconds).

• Frequency - Number of complete cycles

in a period of time (Hz).

• Amplitude – Displacement from the

equilibrium position. It is a measure of the

energy of an oscillator (Different Units).

Definitions

• Equilibrium Position - The center of

motion; the place at which no forces act.

• Displacement - The distance between the

center (equilibrium position) and location

of the wave at any time.

• Restoring Force – The force exerted on

the medium to bring it back to the

equilibrium position.

Some Visual Aids!

Graphical Representation

• Sine curves describe SHM very well!

y = Asin(wt)

Relating Period and Frequency

• The period and the frequency always have

the same definition, regardless of the topic

being discussed.

• Remember, these two values are inverses

when in SI units:

f = 1/T T = 1/f

Example

Terry jumps up and down on a trampoline

with a frequency of 1.5 Hz. What is the

period of Terry’s jumping?

f = 1.5 Hz

T = 1/f = 1/(1.5) Hz

T = .67 s

Springs

• The object to the left is a

common spring system.

• It has a mass of some

kind attached to a spring.

• This spring is stretched

and released. This

causes the entire system

to oscillate.

Springs • The spring supplies the

restoring force on the

mass.

• As the mass gets further

away from the equilibrium

position, the force upon it

gets greater.

• There is no force on the

mass at the equilibrium

position.

Springs

• So the equation for force of a spring is as

follows:

F = kx

(Hooke’s Law)

F – the force supplied by the spring

k – the spring constant (depends on how the

spring is made)

x – displacement of the spring from its

equilibrium position

Example 2

I have a slinky with a spring constant of

100 N/m. If I stretch the slinky 5 meters

from its equilibrium position, with what

force will the spring pull on my hand?

F = kx

F = (100 N/m)(5m) = 500 N

F = 500 N

Period

• If we stretch a spring with a mass and

release it, it will oscillate.

This is SHM!

What is the period of this

Motion?

Period

• The period of a spring system is given by

the equation below:

T =

T – the period of motion

m – Mass of the body attached

k – spring contant

km /2

Period

• There are some important things to notice

about this equation:

1. The larger the mass attached to the

spring, the longer the period.

2. The stronger the spring, the shorter the

period.

3. Remember that period is always in

seconds!

Example

• What is the mass of my car if the shocks

have a spring constant of 6000 N/m and it

oscillates with a period of 2 seconds when

I hit a bump in the road?

T =

(T/2π)2 = m/k

(6000 N/m)(2 s/2π)2 = m

607.9 kg = m

km /2

Pendulums

The object on the left is a

pendulum.

It is usually a mass

hanging on the end of a

string.

It could also be a mass on

the end of a pipe or

other variation.

Pendulums

Gravity supplies the

restoring force to create

Simple Harmonic

Motion.

Note that the higher the

pendulum goes, the

more gravity acts to

bring the pendulum back

to its equilibrium

position.

Restoring Force

The restoring force for

a pendulum is the

component of gravity

that is tangent to its

circular motion.

In most cases:

F = mgsin(Θ)

Example

• What restoring force does gravity supply to

a 0.5 kg pendulum that is at 30 degrees?

F = mgsin(Θ)

F = (.5 kg)(10 m/s2)sin(30)

F = 2.5 N

• What is the restoring force at 90 degrees?

F = mgsin(Θ)

F = (.5 kg)(10 m/s2)sin(90)

F = 5 N

Period

• The period of a pendulum is given by the

following formula:

T =

L – Length of the Pendulum

g – Acceleration due to Gravity

T – Period

gL /2

Period

• There are some important things to notice

here:

1. The period does not depend on the

mass of the pendulum bob.

2. The length and gravity are the only

values that affect the period.

3. As the length of a pendulum becomes

longer, the period becomes longer.

Example

• I find an old grandfather clock with a period

of 1.0 s per swing. If the grandfather clock

is located in Montgomery (g = 10 m/s2),

how long is the pendulum?

T =

(T/(2π))2 = L/g

g(T/(2π))2 = L

(10 m/s2)(1s/(2π))2 = L

0.25m

gL /2