Advanced Higher Physics Unit 1

Simple Harmonic Motion(SHM)

OscillationAn oscillation is a regular to and fro movement around a zero point.For example, a simple pendulum or a mass on a spring (horizontally or Vertically).

zero point

zero point

Examples of SHM

For each example of SHM, state how Ep is stored and where Ek is coming from:

Example Ep stored as: Ek coming from:

Pendulum

Mass on a spring

Trolley moving between springs

Cork floating

Potential energy of mass

Elastic energy of spring

Elastic energy of springs

Potential energy of cork

Moving mass

Moving mass

Moving trolley

Moving cork

Period and frequency

Tw

2

An oscillation has a:

•period (T) which is the time taken for one complete cycle

•Frequency (f) which is the number of cycles made per second.

fw 2

Not in data booklet. In data booklet.

fT

1

In data booklet(higher part)

Link between SHM and circular motion

Circular motion SHM

turntable shadow

Restoring force

For all oscillations there is an unbalanced force acting on the object, and this force is opposite to the motion – so it is called a restoring force.

If this restoring force is proportional to the displacement from thezero position then the oscillation is called Simple Harmonic Motion.

Fun=0N

F

F

weight

tension

Zero position

Simple Harmonic motion

kya

ya

yF

2

2

dt

yda

Where y is the displacement from the zero position

Since F=ma and m is constant.

However as

ywdt

yd 22

2

and k is defined as w² for SHM

In data booklet

ym

k

dt

yd

kydt

ydmmaF

2

2

2

2

The constant is related to the period of the motion by m

km

kw 2

Therefore ywdt

yd 22

2

In data booklet

Variation of period of oscillation with mass for a spring

Solution to the SHM equation

ywdt

yd 22

2

wtAy cos

The expression for y which fits requires

advanced calculus so we will limit ourselves to proving given solutions.

wtAy sin if y=0 at t=0

if y=A at t=0

In data booklet

In data booklet

With A maximum amplitude of Oscillation (from zero position)

A Zeroposition

Proof 1

wtAy sin

wtAwdt

dycos

wtAwdt

ydsin2

2

2

ywdt

yd 22

2

See data booklet for derivatives.

You need to be able to derive this!

Proof 2

wtAy cos

wtAwdt

dysin

wtAwdt

ydcos2

2

2

ywdt

yd 22

2

See data booklet for derivatives.

You need to be able to derive this!

Simple pendulum

L

y

yL

ga

θL

y

From s=rθ (circular motion) y=Lθ

with θ in radians.

Hence:

mg

θ

mgcosθ

T

mgsinθ

The restoring force is mgsinθ.Since F=ma, ma=-mgsinθ (as F and θ in opposite directions)

a=-gsinθ a=-gθ (if θ small)

(as ) L

y

As and for SHMyL

g

dt

yd

2

2

ywdt

yd 22

2

henceL

gw 2

HoweverT

w2

henceg

LT 2

You don’t need to be able to derive this!

Determination of “g”

g

LT 2

Velocity at a given time

wtAy sin

wtAwdt

dyv cos

A

ywt sin

Consider the solution

Then

Rearranging these equations gives

Aw

vwt cosand

Then: wtwtA

y

Aw

v 2222

sincos

12

2

22

2

A

y

wA

v

2

2

22

2

1A

y

wA

v

22222 ywwAv

22 yAwv

)( 2222 yAwv

See additional relationships in data booklet

You need to be able to derive this!

In data booklet.

Graphing SHMy

t

A

A

Zero position

0

-A

Graphing SHMy

t

A

A

Zero position

0

Graphing SHM v

A

Zero position

y

t

A

0

v

t0

Graphing v

A

Zero position

y

t

A

0

v

t0

Maximum velocity is at y=0.

Graphing a

A

Zero position

a

t0

ywa 2y

t

A

0

Graphing a

A

Zero position

a

t0

Maximum acceleration is at y=A but is opposite in direction.

ywa 2y

t

A

0

Energy and SHM

2

2

1mvEk

222

2

1yAwmEk

For mass shown:

)(2

1 222 yAmwEk In data booklet

A Zeroposition

Assuming no energy loss, the total energy in the system is equal to the maximum kinetic energy.

This is when v is at a maximum; when y=0.

So the total energy is 22

2

1Amw

Therefore, at displacement y the potential energy is:

ktotalp EEE

22222

2

1

2

1yAmwAmwEp

22

2

1ymwEp In data

booklet

Damping

t

y

In reality, SHM oscillations come to rest because energy is transferred from the system and so the amplitude of the oscillationdrops.This is called DAMPING.

Critical damping

t

y

If the energy transfer is such that there is no displacement pastthe zero point (assuming it starts at A) then the system is said to beCRITICALLY DAMPED.

However, it is possible to overdamp the system (with very large resistance) which means that the time it takes to stop moving isactually larger than if there was no artificial damping and thesystem is allowed to oscillate to rest.