Physics 123

11. Vibrations and Waves

11.1 Simple Harmonic Motion

11.2 Energy in SHM

11.3 Period and sinusoidal nature of SHM

11.4 The Simple Pendulum

11.6 Resonance

11.7/8 Wave motion and types of waves

11.11 Reflection and Interference of Waves

11.12 Standing Waves

Simple Harmonic Motion

Amplitude

The amplitude of this periodic motion is the distance between all of the following except

1. A and F

2. A and B

3. D and F

4. B and F

Amplitude

Amplitude is the maximum displacement from the equilibrium position.

All choices are measures of amplitude except D which is twice the amplitude.

The correct answer is D

Terms of Endearment

• A complete round trip is called an oscillation.

• The period is the time to execute an oscillation. We use the symbol T to denote the period.

• The frequency is the number of oscillations per second. We use the symbol f to denote frequency.

T = 1/f

SHM governed by F = -kx

•The equilibrium position represents the relaxed length of the spring.

•The spring has a tendency to return to this position if it is stretched or compressed.

•The restoring force is given by Hooke's Law: F = - kx

•Periodic Motion governed by F = -kx is called Simple Harmonic Motion (SHM)

Energy in SHM

• The potential energy stored in the spring is

1/2 kx2

• The kinetic energy of the mass is

1/2 mv2

• The total energy is the sum of the KE of the mass and the PE of the spring.

KE + PE is constant

Problem 11.1 . . . Speed of SHM

A 100 g mass is attached to a spring whose spring constant k = 50 N/m. The amplitude

of oscillation A = 5 cm. The maximum speed of the mass , v0 , in m/s, is most nearly

A. 0.1

B. 1

C. 10

D. 100

Solution 11.1 . . . Speed of SHM

KE = PE

Equating the two gives 1/2 mv02 = 1/2 kA2 so

v0 = A (k/m)1/2

Plugging in the values gives v0 = 1 m/s

Correct choice is B

Which graph represents position vs time?

Which graph represents position vs time?

Either A or D

If time starts when displacement is maximum then A and x = A cos 2 f t

If time starts when displacement is zero (equilibrium) then D and x = A sin 2 f t

Which graph represents speed vs time?

Which graph represents speed vs time?

Either A or D

If time starts when displacement is maximum then D and v = v0 sin 2 f t

If time starts when displacement is zero (equilibrium) then A and v = v0 cos 2 f t

Note: Velocity and position graphs are out of step (out of phase) by 900 or /2

Extreme calculus says (trust me!) . . .

v0 = 2 f A

Also we know that v0 = A (k/m)1/2

Put two and two together:

f = (k/m)1/2/ 2

Which graph represents acceleration vs time?

Which graph represents acceleration vs time?

Suppose the position is given by x = A cos 2 f t

We know that F = -kx

So a = - k x /m

So a = - (kA /m) cos 2 f t

Note: Graph of a vs t will look like the one for x vs t except for the negative sign

Simple Pendulum

Period of a Pendulum

The formula for the period of a simple pendulum is

A. T = 2 (m/L)1/2 B. T = 2 (L/g)1/2 C. T = 2 (g/L)1/2 D. T = 2 (A/m)1/2

Period of a Pendulum

We know that the formula for any SHM is: T = 2 (m/k)1/2

The question is what is the "k" referring to in the absence of any spring? The figure indicates that for a simple pendulum our k = mg/L. So

T = 2 (L/g)1/2

Problem 11.2 . . . What about Bob?

Pendulum 1 is 60 cm long and the mass of the bob is 10 g. Pendulum 2 is also 60 cm long but the mass of its bob is 20 g. Which pendulum oscillates faster (higher frequency (f) or smaller time period (T)?

Solution 11.2 . . . What about Bob?

Both have the same frequency. T does not depend on the mass of the bob!

What is a Wave?

A wave is the propagation of a disturbance

Making Waves

1. Pluck a string. A pulse (disturbance) travels down the string.

2. Ripples move outward in water

Note: The disturbance travels but the medium simply oscillates back-and-forth (SHM)

Three Waves!

A wave is crest to crest or trough to trough

Wavelength =

v = / T

v = f

What does the speed of waves depend on?

The speed depends on the properties of the medium. Wave speed in water is different than the speed on a string. Also the type of string and the tension matter

Speed of waves on a string

v = [Tension / Linear Mass Density]1/2

v = [T / (m/L)]1/2

Types of Waves

Transverse Waves:The disturbance travels in a direction perpendicular to the back-and-forth SHM motion of the particles of the

medium (string,water)

Longitudinal Waves:The disturbance travels in the same direction as the back-and-forth SHM motion of the particles of the medium

(sound)

Reflection of a Wave

Reflection: When a wave runs into an obstacle it bounces back.

Example: Echo is the refection of sound waves

Resonance

Resonance: If an external force is applied at the same frequency as the natural frequency the oscillations increase in amplitude.

Example 1: Pushing a child on a swing

Example 2: Tacoma Narrows Bridge Collapse

Example 3: Caruso and the shattered wine glass

Interference

Constructive Interference: When two waves run into each other in step (in phase). The outcome is increased amplitude

Destructive Interference: When two waves run into each other out of step (out of phase). The outcome is decreased amplitude

Standing Waves on a String

A combination of reflection, interference, and resonance makes standing waves on a string!

Standing Waves on a String

Node

AntinodeNode

Node

Antinode

= 2L/3

= L

= 2LL

Problem 11.3 . . . Standing Waves

A vibrator excites a string at a fixed frequency. By using different weights we put the string under different amounts of tension. This makes the string oscillate with different wavelengths (different number of loops). What is the equation relating T and

Solution 11.3 . . . Standing Waves

We know these two equations:

v = f

v = [T / (m/L)]1/2

So f= [T / (m/L)]1/2

= (1/f ) [T / (m/L)]1/2

What does = (1/f ) [T / (m/L)]1/2 mean?

It means that …..

1. The wavelength is proportional to the square root of the tension in the string

2. A graph of vs T1/2 will be a straight line

3. The slope of the line will be 1 / [f (m/L)1/2]

That’s all folks!