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