PHYS16 – Lecture 32

Ch. 15 Oscillations

Oscillations pre-question

• Is a bouncing ball an example of simple harmonic motion?

A) YesB) No

Oscillations pre-question

• Two kids are swinging on two swings of the same height – one kid is a little chubbier than the other. Neglecting frictional forces, which kid completes a back and forth swing in the fastest time?

A) The chubby kidB) The skinny kidC) The kid who pushes off the ground the bestD) Both complete in the same time

Outline for Oscillations

• Simple Harmonic Motion– Position, Velocity, Acceleration– Force– Energy

• Resonance and Damping

Oscillations and Periodic Motion

http://img.tfd.com/ggse/c9/gsed_0001_0012_0_img2993.png

Simple Harmonic Motion

1) About Equil.2) Periodic3) Sinusoidal

Discussion: Examples of SHM?

• Mass on a Spring• Pendulum

• Snowboarder in halfpipe• Bungee jumper• Child on swing• Bobble head doll?

Simple Harmonic Motion:Position, Velocity, Acceleration

Simple Harmonic Motion

T

tAx

2

)sin(

Equilibrium Point

http://www.gailruby.com/Picture4.png

Discussion: Phase

• Your book says that displacement is in terms of cosine and I just said that it is in terms of sine? Who is right?

)cos()90sin()sin(

tAtAxtAx

There is also a phase term (φ) that let’s you set the initial condition.If the oscillation starts at A it is a sine with a φ=90 degrees or a cosinewith a φ=0 degrees.

SHM Position, Velocity, Acceleration

http://www.tutornext.com/system/files/u89/Chapter%2011-3.gif

)sin(

)cos(

)sin(

2 tAa

tAv

tAx

Discussion

• What is the max speed and when does it occur?

Equilibrium Point

/2at t 0 0at t max

v

Av

Example Question

• An object undergoes simple harmonic motion. If the amplitude and period are doubled, the object’s max speed is:

A) QuadrupledB) DoubledC) UnchangedD) HalvedE) Quartered

Simple Harmonic Motion:Restoring Force

Restoring Force

• Force that always points back to the equilibrium position – Example = spring

k is just a constant – for a spring it is the spring constant

xkxF ˆ

http://www.cs.wright.edu/~jslater/SDTCOutreachWebsite/images/gif/spring_mass_dia.gif

Restoring Force – Spring

mkf

mk

xmkx

tAmtAk

xkxF

21 ,

)sin()sin(dtdmma

2

2

2

2

Discussion: Period

• A block on a spring has a period of T. What is the period if:

A) the mass is doubled?

B) k is quadrupled?

C) A is doubled?

T2

T21

T

Restoring Force - Pendulum• What is the period of a pendulum?

http://upload.wikimedia.org/wikipedia/en/thumb/a/a8/Pendulum.png/300px-Pendulum.png

Lg

smallfor ,

)sin(

Lsmgks

mgksF

Simple Harmonic Motion:Energy

Energy in SHM

2

22222

22

21

)(cos21)(sin

21

21

21

kAE

tAktAmE

kxmvE

http://www.farraguttn.com/science/milligan/APPhys/SHMOver_files/image022.jpg

Main Points - SHM

• Movement

• Restoring Force creates oscillation

• Energy is dependent on amplitude

)sin( tAx

2

2

dtxdmkxF

2

21 kAE