Do now: Thursday, April 20, 2023

A mass oscillates with SHM, period T = 2.4 s and amplitude A = 0.12m

1) Use trig to calculate the angle

2) Calculate the time taken to travel from point C to B

3) Calculate the time taken to travel from point B to A

Quickly try: Thursday, April 20, 2023

A mass oscillates with SHM, period T = 2.4 s and amplitude A = 0.12m

4) Where is the mass at t = 1.0s?

5) What is the velocity of the mass at t = 1.0s?

6) What is the acceleration of the mass at t = 1.0s?

More practise: Thursday, April 20, 2023

A mass oscillates with SHM, period T = 2.4 s and amplitude A = 0.12m

7) Where is the mass at t = 1.8s?

8) What is the velocity of the mass at t = 1.8s?

9) What is the acceleration of the mass at t = 1.8s?

44

SHM VelocitySHM Velocity

For For pp and and mm to be to be staying in line…staying in line…

……they must both they must both have the same have the same vertical component vertical component to their speeds at to their speeds at any time.any time.

pm

A

vm vp

What is the relationship between the What is the relationship between the SHM velocity SHM velocity vvmm and the velocity of the and the velocity of the test particle test particle vvpp on the reference circle? on the reference circle?

So substituting this back into the first So substituting this back into the first equation:equation:

SHM VelocitySHM Velocity vvmm must be equal must be equal

to the vertical to the vertical component of component of vvpp::

vmvp

cosmv Aω ωt

cosm pv v The reference circle has The reference circle has

constant constant vvpp::

pv r A

This formula will be given in the exam. Extension: derive using calculusThis formula will be given in the exam. Extension: derive using calculus

66

Displacement-timeDisplacement-time A graph of displacement-time can be A graph of displacement-time can be

drawn by using a rotating radius drawn by using a rotating radius vector known as a vector known as a Phasor.Phasor.

tω t=0

t=1

t=2

t=3

t=5

t=6

t=7

1 72 3 54 6 8

77

Sketch the velocity-time graph Sketch the velocity-time graph for SHMfor SHM

tω t=6

t=7

t=0

t=1

t=3

t=4

t=5

1 72 3 54 6 8

+Aω

– Aω

VelocityWatch video clip

88

Velocity PhasorVelocity Phasor

tω

1 72 3 54 6 8

velocity

displacement

• The length of the velocity phasor is…The length of the velocity phasor is…• What is the phase difference between the velocity phasor What is the phase difference between the velocity phasor

and the diplacement phasor? and the diplacement phasor?

The velocity phasor is 90° (or The velocity phasor is 90° (or /2 rad) /2 rad) aheadahead of the of the displacement phasor displacement phasor

AA

99

SHM AccelerationSHM Acceleration Point Point pp has centripetal has centripetal

acceleration acceleration aapp in in towards the centre of towards the centre of the circlethe circle

The SHM object has The SHM object has acceleration acceleration aam m

towards equilibrium...towards equilibrium...

...which must be ...which must be equal to the vertical equal to the vertical component of component of aapp

pm

am ap

1010

SHM AccelerationSHM Acceleration

am

ap

sinm pa a

Negative sign indicates direction is always

towards equilibrium

2 sinma A t

ButBut a app is centripetal, so: is centripetal, so:

2

pp

va

r 2

A

A

2A

Putting this all together:Putting this all together:

1111

t

t=5

ωt=0

t=6

t=7

t=1

t=2

t=3

1 72 3 54 6 8

Sketch the acceleration-time Sketch the acceleration-time graph for SHMgraph for SHM

1212

All three….All three….

tω

1 72 3 54 6 8

velocity

displacement

acceleration

Clip: Spring SHM with d v and a graphs alongside - 20sec

1313

All three…All three…

If t=0 is not equilibrium, then all the phasors If t=0 is not equilibrium, then all the phasors shift their starting positions.shift their starting positions. The graphs start The graphs start in different positions and the sin/cosine in different positions and the sin/cosine function may change.function may change.

tAa

tAv

tAy

sin

cos

sin

2

Do pg 124 Qu 13-22

1414

AccelerationAcceleration

)definition SHM to this(Compare

then

sin and

sin that gRememberin

2

2

ya

tAa

tAy

m

m