objectives - weebly€¦ · objectives 19.1 characteristics of simple harmonic motion 19.2...

By Liew Sau Poh

1

19. Oscillations

Objectives 19.1 Characteristics of simple harmonic motion 19.2 Kinematics of simple harmonic motion 19.3 Energy in simple harmonic motion 19.4 Systems in simple harmonic motion 19.5 Damped oscillations 19.6 Forced oscillations and Resonance

2

Outcomes a) define simple harmonic motion by means of the

equation a = 2x b) Show that x = xo sin t as a solution of a = 2x c) derive and use the formula v = (A2 x2) d) describe, with graphical illustrations, the variation in

displacement, velocity and acceleration with time e) describe, with graphical illustrations, the variation in

velocity and acceleration with displacement f) derive and use the expressions for kinetic energy and

potential energy g) describe, with graphical illustrations, the variation in

kinetic energy and potential energy with time and displacement

3

Outcomes h) derive and use expressions for the periods of

oscillations for spring-mass and simple pendulum systems

i) describe the changes in amplitude and energy for a damped oscillating system

j) distinguish between under damping, critical damping and over damping

k) distinguish between free oscillations and forced oscillations

l) state the conditions for resonance to occur

4

19.1 Characteristics of SHM This type of motion is the most pervasive motion in the universe.

All atoms oscillate under harmonic motion.

We can model this motion with a linear restoring force.

5

Periodic Motion Motion that repeats in a regular pattern over and over again is called periodic motion.

Simple harmonic motion is a specific type of periodic motion that has a simple sine or cosine wave shape.

6

Position VS. Time graph What is the simple mathematical form of SHM motion? The displacement of the oscillating mass varies sinusoidally as a function of time.

7 8

Periodic Motion Simple Harmonic Motion

Hearbeat Oscillating mass on a Spring

The restoring force of an ideal spring is given by: F = -kx where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in opposite direction to the displacement of the spring. 9

Simple Harmonic Motion When there is a restoring force, F = -kx, simple harmonic motion occurs.

10

19.2 Kinematics of SHM Simple Harmonic Motion (SHM) occurs when the force acting on a body is proportional to the displacement of the body from some equilibrium position (eg. a spring or a pendulum).

11

x=0

Fs = -kx

x Fs = 0

Fs = +kx

-x

19.2 Kinematics of SHM

12

x=0

Fs = -kx

x Fs = 0

Fs = +kx

-x

When the block attached to the spring (left) is displaced a small distance x from equilibrium, the spring exerts a restoring force which is proportional to the displacement:

19.2 Kinematics of SHM

xdt

xdSo

SHMx

tAdt

xda

tAdtdxv

22

2

2

22

2

)(

cos

sin

13

a = - (k/m) x If we try x=A cos(wt+f) as a solution to this equation, we obtain:

Equation for Simple Harmonic Motion

19.3 Energy in SHM Total Energy = Kinetic Energy + Potential Energy E = K + U

14

x k U 2 2

1 v m K 2 2

1

19.3 Energy in SHM

15

txktU

txktU

m

m22

21

22

1

cos

cos22

1 sin txmtK m

txktK m22

21 sin

212 mE k x constant

k m 2

19.3 Energy in SHM Motion

16

turning point

turning point

range of

motion

22

1 xkUconstant2

21

mxkE

KE and PE Conversion

17

222100 kAUKExavAx

02100 2 UkAKEaAvx

222100 kAUKEAavAx

x=0

Fs = -kx

x

Fs = 0

Fs = +kx

-x

Amplitude

18

Amplitude is the magnitude of the maximum displacement.

tAAx coscos

Period, T

19

For any object in simple harmonic motion, the time required to complete one cycle is the period T.

Frequency, f

20

The frequency f of the simple harmonic motion is the number of cycles of the motion per second.

Tf 1

Energy of the Simple Harmonic Oscillator

21 2

222

222

21

cossin.21

cos21

21

kAE

ttAmkmE

PEKEE

tkAkxPE

total

total

total

tAmmvKE 2222 sin21

21

Thus, total energy is proportional to amplitude2.

For a displacement x = A cos (wt+f), we can say that kinetic energy, KE is: Potential energy (elastic) PE is:

Energy transfer

22

KE, U kA2/2

0 t t = 0 corresponds to the stretched spring.

23

KE, U kA2/2

+A -A x

x = 0 corresponds to equilibrium position of spring.

Angular Frequency

24

Since force restoring kx ) ( x m 2

m k T f or

k m dt

x d m

2 1 / 1

/

2

2

19.4 Systems in SHM

25

1. Pendulums 1. The Simple Pendulum 2. The Physical Pendulum 3.

2. SHM & Uniform Circular Motion 3. Damped SHM 4. Forced Oscillations & Resonance

Gravitational Pendulum

26

Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L.

Simple Pendulum

27

a t x t

T

2

2

T Lg

2SHM for small

2LmI

L F L FI

mg LI

g gsin

acceleration ~ - displacement SHM

The Simple Pendulum

28

m

T

mg

L

x

g L T

L g

L g

dt d

L x But dt

x d m mg

)

2

in rad

(sin

sin

2

2

2

2

2

Comparing a =

A pendulum leaving a trail of ink:

29

Physical Pendulum

30

Pivot

Center of Mass

quick method to measure g

A rigid body pivoted about a point other than its center of mass (com). SHM for small

acceleration ~ - displacement SHM

a t x t

T

2

2

T Img h

2

I

m g h I

h F h F g g sin

The Torsional Pendulum

33

Torsion Pendulum:

2 IT

m Ik

2

2

dI Idt

Spring:

Simple Harmonic Motion

34

Any Oscillating System:

2 mTk

T 2 inertiaspringiness

T Lg

2 2 IT

T Img h

2

SHM & Uniform Circular Motion

35

The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM.

The execution of uniform circular motion describes SHM. http://positron.ps.uci.edu/~dkirkby/music/html/demos/SimpleHarmonicMotion/Circula

r.html


36

radius = xm

The reference point

of radius xm. The projection of xm on a diameter of the circle executes SHM.

ta n g l eUC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations

txtx m c o s


37

xm. The projection of xm on a diameter of the circle executes SHM.

x(t) v(t) a(t)

radius = xm mxv mxa 2

txtx m c o stxtv m s in

txta m cos2


38

The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM.

Measurements of the angle between Callisto and Jupiter: Galileo (1610)

earth

planet

Equations of Motion (SHM)

39

a = - 2x [the definition]

x = A cos t

v = - A sin t

a = - 2A cos t

v = ± (A2 - x2 )0.5

Displacement-Time Graph

40

x

t 0

x = A cos t A

-A

Velocity-Time Graph

41

v

t 0

v = A sin t A

A

Acceleration-Time Graph

42

a

t 0

a = A cos t A

A

Velocity-Displacement Graph

43

v v = ± A x )0.5 A

A

A -A t 0

Acceleration-Displacement Graph

44

a a = x [the definition]

A

A

A -A x 0

Phase Relationship

45

0

x

v a

t

Free oscillations When a system oscillates without external forces acting on it, the system is in free oscillation. The amplitude of oscillation is constant, which will not drop.

46

0

Displacement, x

Time

-x0

x0

19.5 Damped Oscillations

47

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


48

Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is

modifies the undamped oscillation.

Damped SHM

49

SHM in which each oscillation is reduced by an external force.

F k x

Damping Force In opposite direction to velocity Does negative work Reduces the mechanical energy

F bvD

Restoring Force SHM

Damped SHM

50

netF m a

k x b v m a

differential equation

2

2

dx d xk x b mdt dt


51

A graph for a damped oscillation The amplitude decreases with time The blue dashed lines represent the envelope of the motion

19.5 Damped Oscillation

52

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 and is called the damping coefficient


53

However, if the damping is large, it no longer resembles SHM at all.

A: underdamping: there are a few small oscillations before the oscillator comes to rest.


54

B: critical damping: this is the fastest way to get to equilibrium.

C: overdamping: the

system is slowed so much that it takes a long time to get to equilibrium.


55

There are systems where damping is unwanted, such as clocks and watches. Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.


56

m d xdt

k x b dxdt

2

20

2nd Order Homogeneous Linear Differential Equation: Solution of Differential Equation:

x t x e tm

bm

t

( ) cos2

where: km

bm

2

24

b = 0 SHM

Damped Oscillations

57

x t x e tm

bm

t

( ) cos2

km

2

12

bm

1 small damping2

bm

bm

critically damped2

1 0 " "

21 0 " "2

b overdam pedm

the natural frequency

Exponential solution to the DE

Auto Shock Absorbers

58

Typical automobile shock absorbers are designed to produce slightly under-damped motion

19.6 Forced Oscillations & Resonance

59

Forced oscillations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system.

If the frequency is the same as the natural frequency, the amplitude becomes quite large. This is called resonance.


60

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


61

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

62

19.6 Forced Oscillations & Resonance The sharpness of the resonant peak depends on the damping. If the damping is small (A), it can be quite sharp; if the damping is larger (B), it is less sharp.

Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.

External frequency f


63

When the frequency of the driving force is near to 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


64

Each oscillation is driven by an external force to maintain motion in the presence of damping:

F td0 cos

wd = driving frequency


65

Each oscillation is driven by an external force to maintain motion in the presence of damping.

2nd Order Inhomogeneous Linear Differential Equation:

md xdt

k x m dxdt

F td

2

22

0 cos

km


66

w = natural frequency wd = driving frequency

2nd Order Homogeneous Linear Differential Equation:

Steady-State Solution of Differential Equation:

x t x tm( ) coswhere: x F

m b

bm

m

d d

d

d

0

2 2 2 2 2 2

2 2tan

m d xdt

k x m dxdt

F td

2

22

0 cos

km


67

w = natural frequency wd = driving frequency When w = wd resonance occurs!

km

The natural frequency, w, is the frequency of oscillation when there is no external driving force or damping.

less damping

more damping

x F

m bm

d d

0

2 2 2 2 2 2

68

69

Stop the SHM caused by winds on a high-rise building

70

The weight is forced to oscillate at the same frequency as the building but 1190 degrees out of phase.

400 ton weight mounted on a spring on a high floor of the Citicorp building in New York.

Summary

72

OSCILLATION

Free Damped Forced Oscillations and Resonance

0

Displacement, x

Time -x0

x0

less damping

more damping

objectives - weebly€¦ · objectives 19.1 characteristics of simple harmonic motion 19.2...

Documents