Periodic MotionPeriodic Motion 11

Chapter 15Chapter 15

Oscillatory MotionOscillatory Motion

April 17April 17thth, 2006, 2006

Periodic MotionPeriodic Motion 22

The last steps …The last steps …

If you need to, file your taxes TODAY!If you need to, file your taxes TODAY!– Due at midnight.Due at midnight.

This weekThis week– Monday & Wednesday – OscillationsMonday & Wednesday – Oscillations– Friday – Review problems from earlier in Friday – Review problems from earlier in

the semesterthe semester Next Week Next Week

– Monday – Complete review.Monday – Complete review.

Periodic MotionPeriodic Motion 33

The The FINAL EXAMFINAL EXAM

Will contain 8-10 problems. One will Will contain 8-10 problems. One will probably be a collection of multiple probably be a collection of multiple choice questions.choice questions.

Problems will be similar to WebAssign Problems will be similar to WebAssign problems but only some of the actual problems but only some of the actual WebAssign problems will be on the WebAssign problems will be on the exam.exam.

You have 3 hours for the examination.You have 3 hours for the examination. SCHEDULE: MONDAY, MAY 1 @ 10:00 AMSCHEDULE: MONDAY, MAY 1 @ 10:00 AM

Periodic MotionPeriodic Motion 44

Things that Bounce Things that Bounce AroundAround

Periodic Motion 5

The Simple PendulumThe Simple Pendulum

0

1

)sin(

2

2

2

22

L

g

dt

d

dt

dmLLmg

I

Periodic Motion 6

The SpringThe Spring

02

2

2

2

xm

k

dt

xd

dt

xdmkx

maF

Periodic Motion 7

Periodic Motion From our observations, the motion of these

objects regularly repeats The objects seem t0 return to a given position

after a fixed time interval A special kind of periodic motion occurs in

mechanical systems when the force acting on the object is proportional to the position of the object relative to some equilibrium position If the force is always directed toward the

equilibrium position, the motion is called simple harmonic motion

Periodic Motion 8

The Spring … for a moment Let’s consider its motion at each

point. What is it doing?

Position Velocity Acceleration

Periodic Motion 9

Motion of a Spring-Mass System A block of mass m is

attached to a spring, the block is free to move on a frictionless horizontal surface

When the spring is neither stretched nor compressed, the block is at the equilibrium position

x = 0

Periodic Motion 10

More About Restoring Force The block is

displaced to the right of x = 0 The position is

positive The restoring

force is directed to the left

Periodic Motion 11

More About Restoring Force, 2 The block is at the

equilibrium position x = 0

The spring is neither stretched nor compressed

The force is 0

Periodic Motion 12

More About Restoring Force, 3 The block is

displaced to the left of x = 0 The position is

negative The restoring

force is directed to the right

Periodic Motion 13

Acceleration, cont. The acceleration is proportional to the

displacement of the block The direction of the acceleration is opposite

the direction of the displacement from equilibrium

An object moves with simple harmonic motion whenever its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium

Periodic Motion 14

Acceleration, final The acceleration is not constant

Therefore, the kinematic equations cannot be applied

If the block is released from some position x = A, then the initial acceleration is –kA/m

When the block passes through the equilibrium position, a = 0

The block continues to x = -A where its acceleration is +kA/m

Periodic Motion 15

Motion of the Block The block continues to oscillate

between –A and +A These are turning points of the

motion The force is conservative In the absence of friction, the

motion will continue forever Real systems are generally subject to

friction, so they do not actually oscillate forever

Periodic Motion 16

The Motion

Periodic Motion 17

Vertical Spring

Equilibrium Point

Periodic Motion 18

Ye Olde Math

02

2

xm

k

dt

xd0

2

2

L

g

dt

d

022

2

qdt

qd

Periodic Motion 19

022

2

qdt

qd

)cos(

:

0 tqq

Solution

q is either the displacement of the spring

(x) or the angle from equilibrium (). q is MAXIMUM at t=0 q is PERIODIC, always returning to its

starting position after some time T called the PERIOD.

Periodic Motion 20

Example – the Spring

m

kf

k

mT

tm

kTt

m

k

m

k

tm

kxx

xm

k

dt

xd

2

1

2

2)()(

so same, thestaysfunction T,tWhen t

sin

0

2

0

2

2

Periodic Motion 21

Example – the Spring

L

gf

g

LT

tL

gTt

L

g

L

g

tL

g

L

g

dt

d

2

1

2

2)()(

so same, thestaysfunction T,tWhen t

sin

0

2

0

2

2

Periodic Motion 22

Simple Harmonic Motion – Graphical Representation A solution is x(t)

= A cos (t + A, are all

constants A cosine curve

can be used to give physical significance to these constants

Periodic Motion 23

Simple Harmonic Motion – Definitions A is the amplitude of the motion

This is the maximum position of the particle in either the positive or negative direction

is called the angular frequency Units are rad/s

is the phase constant or the initial phase angle

Periodic Motion 24

Motion Equations for Simple Harmonic Motion

Remember, simple harmonic motion is not uniformly accelerated motion

22

2

( ) cos ( )

sin ( t )

cos( t )

x t A t

dxv A

dt

d xa A

dt

Periodic Motion 25

Maximum Values of v and a Because the sine and cosine

functions oscillate between 1, we can easily find the maximum values of velocity and acceleration for an object in SHM

max

2max

kv A A

mk

a A Am

Periodic Motion 26

Graphs The graphs show:

(a) displacement as a function of time

(b) velocity as a function of time

(c ) acceleration as a function of time

The velocity is 90o out of phase with the displacement and the acceleration is 180o out of phase with the displacement

Periodic Motion 27

SHM Example 1

Initial conditions at t = 0 are x (0)= A v (0) = 0

This means = 0 The acceleration

reaches extremes of 2A

The velocity reaches extremes of A

Periodic Motion 28

SHM Example 2 Initial conditions at

t = 0 are x (0)=0 v (0) = vi

This means = /2 The graph is shifted

one-quarter cycle to the right compared to the graph of x (0) = A

Periodic Motion 29

Energy of the SHM Oscillator Assume a spring-mass system is moving

on a frictionless surface This tells us the total energy is constant The kinetic energy can be found by

K = ½ mv 2 = ½ m2 A2 sin2 (t + ) The elastic potential energy can be found

by U = ½ kx 2 = ½ kA2 cos2 (t + )

The total energy is K + U = ½ kA 2

Periodic Motion 30

Energy of the SHM Oscillator, cont The total mechanical

energy is constant The total mechanical

energy is proportional to the square of the amplitude

Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the block

Periodic Motion 31

As the motion continues, the exchange of energy also continues

Energy can be used to find the velocity

Energy of the SHM Oscillator, cont

2 2

2 2 2

kv A x

m

A x

Periodic Motion 32

Energy in SHM, summary

Periodic Motion 33

SHM and Circular Motion This is an overhead

view of a device that shows the relationship between SHM and circular motion

As the ball rotates with constant angular velocity, its shadow moves back and forth in simple harmonic motion

Periodic Motion 34

SHM and Circular Motion, 2 The circle is

called a reference circle

Line OP makes an angle with the x axis at t = 0

Take P at t = 0 as the reference position

Periodic Motion 35

SHM and Circular Motion, 3 The particle moves

along the circle with constant angular velocity

OP makes an angle with the x axis

At some time, the angle between OP and the x axis will be t +

Periodic Motion 36

SHM and Circular Motion, 4 The points P and Q always have the

same x coordinate x (t) = A cos (t + ) This shows that point Q moves with

simple harmonic motion along the x axis

Point Q moves between the limits A

Periodic Motion 37

SHM and Circular Motion, 5 The x component

of the velocity of P equals the velocity of Q

These velocities are v = -A sin (t +

)

Periodic Motion 38

SHM and Circular Motion, 6 The acceleration of

point P on the reference circle is directed radially inward

P ’s acceleration is a = 2A

The x component is –2 A cos (t + )

This is also the acceleration of point Q along the x axis

Periodic Motion 39

SHM and Circular Motion, Summary Simple Harmonic Motion along a straight

line can be represented by the projection of uniform circular motion along the diameter of a reference circle

Uniform circular motion can be considered a combination of two simple harmonic motions One along the x-axis The other along the y-axis The two differ in phase by 90o

Periodic Motion 40

Simple Pendulum, Summary The period and frequency of a

simple pendulum depend only on the length of the string and the acceleration due to gravity

The period is independent of the mass

All simple pendula that are of equal length and are at the same location oscillate with the same period

Periodic Motion 41

Damped Oscillations In many real systems,

nonconservative forces are present This is no longer an ideal system (the type

we have dealt with so far) Friction is a common nonconservative

force In this case, the mechanical energy of

the system diminishes in time, the motion is said to be damped

Periodic Motion 42

Damped Oscillations, cont A graph for a

damped oscillation The amplitude

decreases with time

The blue dashed lines represent the envelope of the motion

Periodic Motion 43

Damped Oscillation, Example One example of damped

motion occurs when an object is attached to a spring and submerged in a viscous liquid

The retarding force can be expressed as R = - b v where b is a constant b is called the damping

coefficient

Periodic Motion 44

Damping Oscillation, Example Part 2 The restoring force is – kx From Newton’s Second Law

Fx = -k x – bvx = max

When the retarding force is small compared to the maximum restoring force we can determine the expression for x This occurs when b is small

Periodic Motion 45

Damping Oscillation, Example, Part 3 The position can be described by

The angular frequency will be

2 cos( )bt

mx Ae t

2

2

k b

m m

Periodic Motion 46

Damping Oscillation, Example Summary When the retarding force is small, the

oscillatory character of the motion is preserved, but the amplitude decreases exponentially with time

The motion ultimately ceases Another form for the angular frequency

where 0 is the angular frequency in the absence of the retarding force

220 2

b

m

Periodic Motion 47

Types of Damping

is also called the natural frequency of the

system If Rmax = bvmax < kA, the system is said to

be underdamped When b reaches a critical value bc such that

bc / 2 m = 0 , the system will not oscillate The system is said to be critically damped

If Rmax = bvmax > kA and b/2m > 0, the system is said to be overdamped

0

k

m

Periodic Motion 48

Types of Damping, cont Graphs of position

versus time for (a) an underdamped

oscillator (b) a critically

damped oscillator (c) an overdamped

oscillator For critically damped

and overdamped there is no angular frequency

Periodic Motion 49

Forced Oscillations It is possible to compensate for the

loss of energy in a damped system by applying an external force

The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces

Periodic Motion 50

Forced Oscillations, 2 After a driving force on an initially

stationary object begins to act, the amplitude of the oscillation will increase

After a sufficiently long period of time, Edriving = Elost to internal Then a steady-state condition is reached The oscillations will proceed with constant

amplitude

Periodic Motion 51

Forced Oscillations, 3 The amplitude of a driven oscillation

is

0 is the natural frequency of the undamped oscillator

0

222 2

0

FmA

bm

Periodic Motion 52

Resonance When the frequency of the driving force

is near the natural frequency () an increase in amplitude occurs

This dramatic increase in the amplitude is called resonance

The natural frequency is also called the resonance frequency of the system

Periodic Motion 53

Resonance At resonance, the applied force is in

phase with the velocity and the power transferred to the oscillator is a maximum The applied force and v are both

proportional to sin (t + ) The power delivered is F . v

This is a maximum when F and v are in phase

Periodic Motion 54

Resonance Resonance (maximum

peak) occurs when driving frequency equals the natural frequency

The amplitude increases with decreased damping

The curve broadens as the damping increases

The shape of the resonance curve depends on b

Periodic Motion 55

WE ARE DONE!!!