PHYS16 – Lecture 32

Simple Harmonic Motion November 29, 2010

Exam Answers

Question 1 – Lift of Plane

• Use Bernoulli to calculate lift

• Use Newton’s laws

MN 28.645 ),(

2 ),(2 22

wingone

F

vvvvPPAF bottopbottop

MN 25.4

)sin()sin()sin(

FdtdVv

dtdmvmv

dtd

dtdpF

Question 2 – Height of sky and ocean

• Height of sky given constant density

• Height of sky given that density changes

• Height of ocean given constant density

km 9.7

gPh

km 18 ,00 /

0

hePP Pgh

km 1.11

gPh

Question 3 – Car performance

• Power of car

• Torque of car (4 wheels)

kW 220

))sin(

P

mgFFFvP

friction

Nm 2600or /

FrP

Question 3 – Car performance

• Rpm of engine

• Torque on wheelsrpm 11000 1st rpm, 6300 2nd

rpm 42003rd rpm, 32004th rpm, 2400 5th

wheeldiffgearengine GRGR

Nm 4500 1st Nm, 6002 2ndNm 14003rd Nm, 8804th Nm, 550 5th

enginediffgearwheels GRGR

Question 4 – Answer is 42

• Time for ball to tunnel through Earth

• Distance to geostationary orbit

min 424

334

E

E

Gt

mrGF

Earth ofcenter thefrom Mm 42

4 2

3

2

rGMr

T

E

Question 4 – Answer is 42

• Time for low orbit satellite to go ½ around Earth

• Height of object on Ganymede

km 4221

281

2

hhR

GMR

GMv

RGMv

initial

initial

min 422

4 2

3

2

TtGMR

T

EE

Question 5 – Swing set

• John jumps on Merry-go-round

• Balancing multiperson see-saw

m 6-at kg 50 and 2m,at kg 35 m, 4at kg 20 m, 6at kg 25044332211 xmxmxmxm

rad/s 33.2

Johnmerry

JohnmerryJohnmerrymerryJohnmerrymerry IrpI

Question 5 – Swing set

• Mass of kid on tippy bridge

kg 5.250)sin()cos(

0)cos()sin(

0

21

21

2

mTT

TTmgMgTplank

Outline for Oscillations

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

• Waves– Mechanical waves– Intensity and Power– Superposition and Interference– Standing Waves

Simple Harmonic Motion:Position, Velocity, Acceleration

Oscillations and Periodic Motion

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

Simple Harmonic Motion

T

tAx

2

)sin(

Equilibrium Point

1) About Equilibrium2) Periodic3) Sinusoidal

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

Examples

• Mass on a Spring• Pendulum

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

SHM Position, Velocity, Acceleration

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

)sin(

)cos(

)sin(

2 tAa

tAv

tAx

Example Question

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

Equilibrium Point

Tat 0T/2at max

vAv

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

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

SHM – Restoring Force

mkf

mk

xmkx

tAmtAk

xkxF

21 ,

)sin()sin(dtdmma

2

2

2

2

Example Question

• 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

Example Question• 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