Simple Harmonic Motion - Owen Shepherd

Simple harmonic motion definition

Where the acceleration of an object is directly proportional to its displacement, IN THE OPPOSITE DIRECTION

Repeatedly acts to restore an object to its equilibrium

If it’s damped, it will

Formula for SHM

a = -w²x

a is proportional to the negative of x, through w

w = 2pif

How does this work?

As the frequency remains constant throughout

W remains constant

So at minimum displacement?

Acceleration =

Velocity =

Acceleration = zero

Velocity = max

At max displacement

Acceleration =

Velocity =

Acceleration = max opposite to displacement

Velocity = 0

Time period

Remains constant in SHM

What graph does displacement form

Sin graph

Due to constant changing rates of acceleration opposite to it

How to get to velocity graph from displacement

Differentiate!!!

Gradient of the graph

This means that relationships displacement to velocity to acceleration?

Displacement is sin

Velocity is differentiated sin

Acceleration is twice differentiated sin = which is just -sin = SHM!

To have an acceleration there must be?

A force constantly to the centre

How to Calc force from a

F = ma

Times a by m of object

F = -w²xm

Where does V max occur?

At 0 displacement

So v max = wroota²

As can exclude x from equation of or v on equation sheet

Mass spring systems

Mass hanging from a spring

Exhibits SHM as it passes through an equilibrium

And obeys HOOKES LAW

How does obeying hookes law make it SHM

F = ke

Force is opposite to resolving force

So F = -ke

Next step to prove SHM

Convert force into a

F / m = a

-ke / m = a

Extension is actually just?

Displacement

So now a =

-kx/ m

So compare the two

a = -w²x

a = -k/m x

So w² =

k / m

So w =

Root k / m

2pif = root k/m

So gives the mass spring system equation when resolve for T

What so I need to know?

That hookes law, once the force converted into an acceleration, shows that a mass spring system exhibits SHM

Pendulums

Exhibit SHM

Due to the small angle approximation

Force that returns pendulum to centre?

Component of the resultant on the weight

Displaced horizontally, so connecting string pulls horizontally

Resolving force in pendulum = ?

Mgsin

Angle between supporting rope and centre

So acceleration in pendulum system =

Gsin

Small angle approximation does what?

Removes sin

Convert sin to just theta or tan theta

This then allows

Creation of pendulum equation

Don’t need to know marsh in between

Need to know about pendulum system

Can only be considered SHM if deflection below 10 degrees

As small angle approximation that defines it only applies here

Energy when compared to displacement

Energy is a?

Scalar

So has two peaks for a full oscillation

On forwards ad one backwards

As no direction conveyed on graph

Energy is also?

Opposite to displacement

As displacement returns to 0

Energy is at a max

As velocity is at a max

Gradient of displacement

Why is it two peaks for one oscillation

Scalar quantity

So when pass through centre it peaks

Pass through once on way out, and once on way back

Damping

Energy is removed from a SHM system

Reduces the amplitude of oscillations

When does damping occur

When energy is transferred to the environment

Light damping

Reduces the amplitude by a fractional amount each oscillation

Heavy damping

Amplitude gets much smaller after an oscialation

Critical damping

Returns the system to its equilibrium as fast as possible

Without passing the equilibrium

Critical damping use

Car suspension

Overdamping

Goes back to equilibrium eventually

Much slower than naturally

Slows the oscillation

Doesn’t go past

Uses of overdamping

Heavy doors

Resonance

When the frequency applied to an object matches it natural frequency

Leads to maximum energy transfer and maximum amplitude oscillations

Free vibration

Object oscillates at its natural frequency with no external driving force

Forced vibration

Object oscillates at a frequency which it is driven at by an external periodic driving force

When free and forced match

RESONSNACE!

Amplitude vs frequency

Impact on this graph when damped

Graph shifts down and left

Frequency of resonance decreases as mass increased

Highest amplitude reduces as so much energy lost to environment

Resonance phase difference

When of same frequency, but not collide at perfect time

Significantly below 90 degrees = in phase but not resonant

At or around 90 = perfectly in phase and resonant

Above 90 degrees = in anti phase = not resonant

How to measure f of SHM

Use a fiducial marker

Wait for multiple osicallations and take average

Then convert T to f