# Simple Harmonic Motion

Extracts from this document...

Introduction

Simple Harmonic Motion:

What is Simple Harmonic Motion

- It is simple – not many forces
- It is related to periodic waves
- Its about motion (lol)

Example of SHM

- A person in the ocean will experience SHM as waves go past.

Example:

- Hang a mass on a spring
- Pull it down a little and let it go!

In common:

- Oscillations: back and forth, up and down
- Nice and stable: if you push it or pull it and let it go

You will get SHM if:

- The restoring force is directed towards the equilibrium position, i.e. goes back towards where it started
- The restoring force is proportional to the distance from the equilibrium. i.e. pull pendulum a long way, there is a bigger force.

e.g. pull the mass on the spring and the spring stretches and pulls back to where it normally hangs when you let it go.

WAVES

- λ = wavelength (m)
- T = period (s)
- A or x0 = amplitude (m)

- Period is the time it takes for ONE wave, frequency is the number of waves PER second

CIRCLES

- Sine waves
- Just expresses where they start periodically
- f (frequency) is the symbol we use for how often history repeats
- f is measured in hertz, Hz
- f = 1/T or f-1
- e.g. a frequency of 2Hz for a pendulum swinging means it swings twice in a second.

f = 1/T

ώ

- ώ is “omega”
- “angular frequency” or “angular velocity” (the speed it takes to go around a circle)
- think of V = D / T
- in wave motion: ώ = d/t = 2(pi) / T = 2(pi)f
- a sine wave can express a circle, this is just like saying you do one top of a circle (distance 2(pi)) in period (T)
- ώ is really degrees per second.

ώ = 2(pi) / T = 2(pi)f

Radians

- SI unit of an angle
- 2(pi) radians = 360 degrees.

2(pi) radians = 360 degrees

More

- to describe the motion of SHM we can use angles, say, for a pendulum
- measuring the time is easier than angles
- θ = ώt ; angle = degrees (or radians per second) x time

θ = ώt

The Math’s: Position

- if you pull a weight on a string, or a pendulum, you can make SHM
- Distance is ‘x’, amplitude is ‘x0’
- x(t) = Acosθ, but we want to know time so we use θ = ώt
- x(t) = Acos(ώt) or just x = x0cos(ώt)
- swings back and forth from A to –A
- the period (T) is found by 2(pi)/ώ

x = x0cos(ώt)

Velocity

- Just like in mechanics we sometimes want to know the velocity
- Remember velocity = change in displacement / time
- The gradient of a cosine graph is a negative sine graph
- v(t) = -ώAsin(ώt) or just V = -v0sin(ώt)
- where v0 = ώx0 – the maximum velocity (because it is maximum at the highest point)

Middle

a = -ώ2x0

Mass on a Spring

- if you hang a mass on a spring it will stretch the spring a bit and then just sit there
- 2 forces: - weight = mg (down)

- spring = -kx (up)

- “k” is the spring constant: if there is a big number, it is a tight spring, if there is a small number, it is a loose spring
- it tells you how many Newton’s of force you will feel if you pull a spring ‘x’ meters from where it usually is.
- If spring is not moving, the forces are equal at this point

Spring into Action

- F = -kz, where k = mώ2

(Hooke’s Law)

v = -ώ√(x02 – x2)

SHM: Initial Conditions, (boundary conditions)

- Remember that you can make a pendulum swing by

Conclusion

- The waves get smaller
- E.g. a pendulum with some friction or air resistance.
- The amplitude slowly decreases with time.
- However – the period does not change

Heavy (OVER) Damping

- E.g. of a pendulum in water
- Comes to a stop quickly

Critical Damping

- E.g. pendulum in honey
- NO oscillations

SHM: Natural Frequency (Resonant Frequency)

- A person on a swing will swing back and forth at a particular frequency. This is called the natural frequency of the swing.
- Blowing across the top of the bottle makes a sound. That is the natural frequency of the bottle
- A guitar string plucked plays a particular note – its natural frequency
- Every object has a natural frequency of vibration

- E.g. a glass has a natural frequency
- If you tap a glass you hear a sound. The frequency of that sound is the glasses natural frequency
- Normally the sound fades away due to damping. If we introduce some FORCING we can keep it vibrating
- E.g. pushing someone on a swing.
- If you push at the same frequency the swing would naturally have, you get a big swing. This is called RESONANCE.
- Forcing at an objects NATURAL FREQUENCY creates RESONANCEi.e. more bang for your buck.

In a Graph

This student written piece of work is one of many that can be found in our International Baccalaureate Physics section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month