Waves and Harmonic Motion AP Physics M. Blachly. Review: SHO Equation Consider a SHO with a mass of 14 grams: Positions are given in mm.

Waves Waves and and Harmonic Harmonic MotionMotion

AP PhysicsAP Physics

M. BlachlyM. Blachly

Review: SHO Equation

Consider a SHO with a mass of 14 grams:

Positions are given in mm

( ) 25cos(16 ) 10y t t

SHO Equation

Find the following:

T

k

At t = 3, find

y

Etotal

a

Period of a Harmonic Oscillator

2m

Tk

For a system with a mass m, and a linear restoring force characterized by k, the period is:

Pendulum

For small angles, gravity acts as a linear restoring force.

/

s

mgF x

L

k mg L

x

Pendulum

2L

Tg

Pendulum

A 500 g. mass is suspended from a 2 m long string. It is then pulled to a position of x=4 cm and released. Write the equation for the position of this pendulum at any time.

Damping

The damped Harmonic Oscillator is a system where energy is not conserved.Applet: http://www.lon-capa.org/~mmp/applist/damped/d.htm

In the real world, there is always some damping, from friction, air resistance, internal deformations of the medium, etc.

Damped SHO Examples

Car Spring and shocks

• Lots of damping is good

Radio receivers

• Very little damping is good

Wave Motion

Types of Waves:

• Transverse

• Longitudinal

• Torsional

Characteristics of Waves

• Frequency

• Period

• Amplitude

• Wavelength

• velocity

Wave Speed

v f

Wave Phenomena

Superposition

Interference

Beats

Types of Waves

Longitudinal: The medium oscillates in the same direction as the wave is moving

• Sound

• Slinky demo

Transverse: The medium oscillates perpendicular to the direction the wave is moving.

• Water (more or less)

• Slinky demo

8

Slinky TnR

Suppose that a longitudinal wave moves along a Slinky at a speed of 5 m/s. Does one coil of the slinky move through a distance of five meters in one second?

1. Yes

2. No

5m

12

Harmonic Waves

Wavelength

Wavelength: The distance between identical points on the wave.

Amplitude: The maximum displacement A of a point on the wave.

Amplitude A

A20

y(x,t) = A cos(t –kx)

Angular Frequency: = 2 f

x

y

Wave Number k: k = 2 /

Recall: f = v /

Period and Velocity Period: The time T for a point on the wave to undergo one

complete oscillation.

Speed: The wave moves one wavelength in one period T so its speed is v = / T.

Tv

22

TnR

Suppose a periodic wave moves through some medium. If the period of the wave is increased, what happens to the wavelength of the wave assuming the speed of the wave remains the same?

1. The wavelength increases

2. The wavelength remains the same

3. The wavelength decreases

correct

= v T

26

The wavelength of microwaves generated by a microwave oven is about 3 cm. At what frequency do these waves cause the water molecules in my burrito to vibrate ?

(a) 1 GHz (b) 10 GHz (c) 100 GHz

1 GHz = 109 cycles/sec

The speed of light is c = 3x108 m/s

ACT

29

Recall that v = f.

1 GHz = 109 cycles/sec

The speed of light is c = 3x108 m/s

GHz8

10v 3 10 m sf 10 Hz 10

.03m

H H

O

Makes water molecules rotateMakes water molecules rotate

Solution

30

Absorption coefficientof water as a functionof frequency.

f = 10 GHz

Visible

“water hole”

31

Water and Waves

Interference and Superposition

When too waves overlap, the amplitudes add.

• Constructive: increases amplitude

• Destructive: decreases amplitude

34

ReflectionA slinky is connected to a wall at one end. A pulse travels to the right, hits the wall and is reflected back to the left. The reflected wave is

A) Inverted B) Upright

• Fixed boundary reflected wave is inverted

• Free boundary reflected wave upright

37

Standing Waves Fixed EndpointsFundamental n=1

n = 2L/n

fn = n v / (2L)

Wave Speed

Fv

Standing Waves:

f1 = fundamental frequency (lowest possible)

L / 2

A guitar’s E-string has a length of 65 cm and is stretched to a tension of 82N. If it vibrates with a fundamental frequency of 329.63 Hz, what is the mass of the string?

v = f = 2 (0.65 m) (329.63 s-1) = 428.5 m/s

v2 = F / = F / v2

m= F L / v2

= 82 (0.65) / (428.5)2 = 2.9 x 10-4 kg