chapter 15 oscillations. periodic motion periodic (harmonic) motion – self-repeating motion...

Chapter 15

Oscillations

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 15Problem 5

A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz. (a) Show that the position of the particle is given by x = (2.00 cm) sin (3.00 π t). Determine (b) the maximum speed and the earliest time (t > 0) at which the particle has this speed, (c) the maximum acceleration and the earliest time (t > 0) at which the particle has this acceleration, and (d) the total distance traveled between t = 0 and t = 1.00 s.

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

Chapter 15Problem 17

A 50.0-g object connected to a spring with a force constant of 35.0 N/m oscillates on a horizontal, frictionless surface with an amplitude of 4.00 cm. Find (a) the total energy of the system and (b) the speed of the object when the position is 1.00 cm. Find (c) the kinetic energy and (d) the potential energy when the position is 3.00 cm.

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 15Problem 27

A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length R. That is, Rg /

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 15

Problem 2(a) 4.33 cm(b) −5.00 cm/s(c) −17.3 cm/s2

(d) 3.14 s; 5.00 cm

Answers to the even-numbered problems

Chapter 15

Problem 16(a)0.153 J(b) 0.784 m/s(c) 17.5 m/s2

Answers to the even-numbered problems

Chapter 15

Problem 261.42 s; 0.499 m