Chapter 16 Oscillations

Guitar string; drum; diaphragms in telephones andspeaker systems; quartz crystal in wristwatches ….

Oscillations in the real world are usually damped; that is, the motion dies out gradually, transferring mechanical energy to thermal energy by the action of frictional forces. Although we cannot totally eliminate such loss of mechanical energy, we can replenish the energy from some source. For example, by swinging your legs or torso you can “pump” a swing to maintain or increasethe oscillations. In doing this, you transfer biochemical energy to mechanical energy of the oscillating system.

Oscillations motions that repeat themselves. Fundamental physical phenomena

§1 Simple Harmonic Motion

A “snapshots” of a simple oscillating system,a particle moving repeatedly back and forthabout the origin of an x axis.

(a) A sequence of “snapshots” (taken at equal time intervals) showing the position of a particle as it oscillates back and forthabout the origin along an x axis, betweenthe limits +xm and -xm. The vectorarrows are scaled to indicate the speedof the particle.(b) A graph of x as a function of time for the motion of (a).

One important property of oscillatory motion is its frequency, or number of oscillations that are completed each second. The symbol for frequency is f, and its SI unit is the hertz (abbreviated Hz), where

1 hertz=1 Hz=1 oscillation per second=1 s-1.

Related to the frequency is the period T of the motion, which is the time for one complete oscillation (or cycle);that is,

.1f

T

Any motion that repeats itself at regular intervals is called periodic motion or harmonic motion.

A simple harmonic motion is described by followingfunction

).cos( txx m

where xm, , and areconstants. They are basicquantities of a simple harmonic motion.x(t) is a period function,that means x(t) = x(t +T) .For simplicity, we consider=0. Then we have

Ttxtx mm coscos

The cosine function first repeats itself when its argument(the phase) has increased by 2 rad (radian), then we have

that is

2T

,2)( tTt

.22

fT

T=T’xm’>xm

=’=0

T=2T’xm’=xm

=’=0

T=T’xm’=xm

The Velocity of SHM

From the definition

,)cos()(

)( txdtd

dttdx

tv m

.sin txtv m

The Acceleration of SHM

dt

tdvta )( .cos2 txta m

In SHM, the acceleration is proportional to the displacement but opposite in Sign, and the two quantities are related by the square of the angular frequency.

a(t)= -x(t).

Mathematically, the spring force

.kxF

According to Newton’s second law

SHM is the motion executed by a particle of mass m subject to a force that is proportional to displacement of the particle but opposite in sign.

,mk

.2km

T

Sample Problem 16-2 :

§2 Energy in Simple Harmonic Motion

Potential Energy

txm

tkxkxtU

m

m

222

222

cos

)(cos21

21

)(

2

1

φ

Kinetic Energy

txmmvK m2222 sin

21

21

or txkmvK m2222 sin

21

21

Obviously, the total energy is given by

2

21

mkxKUE = constant

Sample Problem:A block of mass M, at rest on a horizontal frictionless table, is attached to a rigid support by a spring of constant k. A bullet of mass m and velocity   strikes the block as shown in the figure . The bulletis embedded in the block. Determine (a) the speed of the block immediately afterthe collision and (b) the amplitude of theresulting simple harmonic motion.

v

Solution:

The problem consists of two distinct parts: the completely inelastic collision (which is assumed to occur instantaneously, the bullet embedding itself in the block before the block moves through significantdistance) followed by simple harmonic motion (of mass m +M attached to a spring of spring constant k).

)/(' Mmvmv (a) Momentum conservation readily yields

(b) Since v’ occurs at the equilibrium position, then v’ = vm for the simple harmonic motion. The relation vm = xm can be used to solve for xm , or we can pursue the alternate (though related) approach of energy conservation. Here we choose the latter:

,21

')(21 22

mkxvMm

.21

)()(

21 2

2

22

mkxMm

vmMm

which simplifies to

.)( Mmk

mvxm

§3 Pendulums

A massless string of length L is fixed at a ceiling. An apple is hanged at the other end, and swings back and forth a small distance.

 

 

The simple Pendulum

Forces acting on the apple are:

force. nalgravitaito the from gm

string, the from

T

The apple swings with a very smallangle . =0 is the so called equilibrium position.

It is easy to see that ;cos TFg

The torque of the system is

,sin Lmg

where the minus sign indicates that the torque reduces.

According to = ImL2is the rotational inertia. Wehave

.sin ImgL

,0sin2

2

Lg

dt

d

Considering that is very small, and sin≈we have

,022

2

dt

d

with The equation is the same as the simple harmonic oscillation.

. LgImgL //)(

TI

mgL2

The Physical Pendulum

A real pendulum, usually called a physical pendulum, can have a complicated distribution of mass, much different from that of a simple pendulum. Does a physical pendulum also undergo SHM? If so, what is its period?

According to the same method as discussed in simple pendulum, one can have

./2 mghIT

A physical pendulum can be used to measure the gravitational acceleration. How to do that?

For a uniform rod, I=(1/3)mL2,and h=L/2. So we have

. ggh

TLL

322

22

3

1

. 2

2

3

8

T

Lg

It is equivalent to a simple pendulum of L0=2L/3.

§4 Simple Harmonic Motion and Uniform Circular Motion

)cos( txx m

Considering various relations of a simple harmonic motion

Position:

Velocity: )sin( txv m

Acceleration: ).cos(2 txa m

You may find that the equations look like to havesome relations with a circular motion. Actually, Galileo first discovered a simple harmonic motion from his observation of four moons of Jupiter, each of them moves forth and back relative to the planet. (see book.)

Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs.

In figure (a), the radium of the circle is xm , the projectionin the x axis of the position vector of point P’ at time t canbe written as follows

).cos( txx m

Similarly, from the figures (b) and (c), we can have thecorresponding projections on x axis are

),sin( txv m

)cos(2 txa mandrespectively.

§5 Damped Simple Harmonic Motion

A block with mass m oscillates vertically on a spring with spring constant k. From the block, a rod extends to a vane (both assumed massless) that is submerged in a liquid. As the vane moves up and down, the liquid exerts an inhibiting drag force on it and thus on the entire oscillating system.

The liquid exerts a damping force d that is proportionalin magnitude to the velocity   of the vane and block (an assumption that is accurate if the vane moves slowly). Then, for components along the x axis, we have

v

,vbFd

where b is a damping constant that depends on the characteristics of both the vane and the liquid and has theSI unit of kilogram per second. The minus sign indicates that    opposes the motion. dF

Then we have a damped harmonic motion,

This equation can be solved analytically and the result is

).cos()()

2(

textxt

m

b

m

We can regard the equation as a cosine function whoseamplitude, whish is , gradually decreases withtime.

mbtmex 2/

where 2

2

4mb

mk

§6 Forced Oscillation and Resonance

For a swing, if someone pushes the swing periodically, the swing has forced, or driven, oscillations. Two angularfrequencies are associated with a system undergoingdriven oscillations: (1) the natural angular frequency of the system, which is the angular frequency at which it would oscillate if it were suddenly disturbed and then left to oscillate freely, and (2) the angular frequency d

of the external driving force causing the driven oscillations.

The general equation is

)(2

2

tFkxdtdx

bdt

xdm

One special solution is

tFtF d cos)( 0

,cos tCx d

where the constant is to be determined. In other words,we might suppose that if we kept pushing back and forth, the mass would follow back and forth in step withthe force. For the case b=0, it is easy to have

. )( 22

0

dm

FC

That is, m oscillates at the same frequency as the force,but with amplitude which depends on the frequency ofthe force, and also upon the frequency of the natural motion of the oscillator.

§7 Superposition of Oscillations

Superposition of Two Simple Harmonic Oscillation in the Same Direction

I. With the same frequencies

)cos(

)cos(

222

111

txx

txx

m

m

)cos()cos( 221121 txtxxxx mm

txx m cos

where

)cos(2 122122

21 mmmmm xxxxx

1

1x

2x

x

x

2211

2211

coscossinsin

tan

mm

mm

xxxx

The resultant oscillation is related to the phase difference2-1 .

)( ,2,1,0,212 kk

We have21 mmm xxx

Then we have maximum oscillation amplitude.

• If

•• If )12(12 k

we have|| 12 mm xxx

II. With two different frequencies

)cos(

)cos(

222

111

txx

txx

m

m

Suppose xm1 = xm2 = A, we have

)2

cos(

)2

cos()2

cos(2

12

121221

tx

ttAxxx

m

)2

cos(2 12 tAxm

Superposition of Two Perpendicular Simple Harmonic Oscillation

)cos(

cos

2

1

tyy

txx

m

m

By eliminating the parameter t,

)(sin)cos(2 122

122

2

2

2

mmmm yx

xy

y

y

x

x

This is an ellipse equation. The path of the particle depends on the phase difference 2 - 1:

(1) 2 - 1=0:

xxy

ym

m

y

x

(2) 2 - 1=

xxy

ym

m

(3) 2 - 1=/2

12

2

2

2

mm y

y

x

x

If xm=ym=A,222 Ayx

Assignments:16 — 9E 16 — 19p16 — 37E