Chapter 13 Oscillatory Motion. Periodic motion Periodic (harmonic) motion – self-repeating motion Oscillation – periodic motion in certain direction Period.

Chapter 13

Oscillatory Motion

Periodic motion

• Periodic (harmonic) motion – self-repeating motion

• Oscillation – periodic motion in certain direction

• Period (T) – a time duration of one oscillation

• Frequency (f) – the number of oscillations per unit time, SI unit of frequency 1/s = Hz (Hertz)

Tf

1

Heinrich Hertz(1857-1894)

Simple harmonic motion

• Simple harmonic motion – motion that repeats itself and the displacement is a sinusoidal function of time

)cos()( tAtx

Amplitude

• Amplitude – the magnitude of the maximum displacement (in either direction)

)cos()( tAtx

Phase

)cos()( tAtx

Phase constant

)cos()( tAtx

Angular frequency

)cos()( tAtx

)(coscos TtAtA 0

)2cos(cos )(cos)2cos( Ttt

T 2

T

2

f 2

Period

)cos()( tAtx

2

T

Velocity of simple harmonic motion

)cos()( tAtx

dt

tdxtv

)()(

)sin()( tAtv

dt

tAd )]cos([

Acceleration of simple harmonic motion

)cos()( tAtx

2

2 )()()(

dt

txd

dt

tdvta

)()( 2 txta

)cos(2 tA

Chapter 13Problem 19

Write expressions for simple harmonic motion (a) with amplitude 10 cm, frequency 5.0 Hz, and maximum displacement at t = 0, and (b) with amplitude 2.5 cm, angular frequency 5.0 s-1, and maximum velocity at t = 0.

The force law for simple harmonic motion

• From the Newton’s Second Law:

• For simple harmonic motion, the force is proportional to the displacement

• Hooke’s law:

maF

kxF

xm 2

m

k

k

mT 22mk

Energy in simple harmonic motion

• Potential energy of a spring:

• Kinetic energy of a mass:

2/)( 2kxtU )(cos)2/( 22 tkA

2/)( 2mvtK )(sin)2/( 222 tAm

)(sin)2/( 22 tkA km 2

Energy in simple harmonic motion

)(sin)2/()(cos)2/( 2222 tkAtkA

)()( tKtU

)(sin)(cos)2/( 222 ttkA

)2/( 2kA )2/( 2kAKUE

Energy in simple harmonic motion

)2/( 2kAKUE

2/2/2/ 222 mvkxkA kmvxA /222

22 xAm

kv 22 xA

Chapter 13Problem 34

A 450-g mass on a spring is oscillating at 1.2 Hz, with total energy 0.51 J. What’s the oscillation amplitude?

Pendulums

• Simple pendulum:

• Restoring torque:

• From the Newton’s Second Law:

• For small angles

)sin( gFL

I

sin

I

mgL

)sin( gFL

Pendulums

• Simple pendulum:

• On the other hand

L

at

I

mgL

L

s s

I

mgLa

)()( 2 txta

I

mgL

Pendulums

• Simple pendulum:

I

mgL 2mLI

2mL

mgL

L

g

g

LT

22

Pendulums

• Physical pendulum:

I

mgh

mgh

IT

22

Chapter 13Problem 28

How long should you make a simple pendulum so its period is (a) 200 ms, (b) 5.0 s, and (c) 2.0 min?

Simple harmonic motion and uniform circular motion

• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

Simple harmonic motion and uniform circular motion

• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

)cos()( tAtx

dt

tdxtvx

)()(

)sin()( tAtvx

Simple harmonic motion and uniform circular motion

• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

dt

tdxtvx

)()(

)sin()( tAtvx

)cos()( tAtx

Simple harmonic motion and uniform circular motion

• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

2

2 )()(

dt

txdtax

)cos()( tAtx

)cos()( 2 tAtax

Damped simple harmonic motion

bvFb Dampingconstant

Dampingforce

Forced oscillations and resonance

• Swinging without outside help – free oscillations

• Swinging with outside help – forced oscillations

• If ωd is a frequency of a driving force, then forced

oscillations can be described by:

• Resonance:

)cos(),/()( tbAtx dd

d

Questions?

Answers to the even-numbered problems

Chapter 13

Problem 200.15 Hz; 6.7 s

Answers to the even-numbered problems

Chapter 13

Problem 3865.8%; 76.4%