Chapter 13

Vibrations and Waves

Simple Harmonic Motion

SHM is oscillating motion in which the position of an object is ‘restored’ by a restoring force.

“Harmonic” refers to harmonic functions of sine and cosine, which can be used to describe the repeating motion.

The simplest example of SHM is that of a mass oscillating at the end of a spring.

Simple Harmonic Motion Hooke’s Law says the

restoring force acting on the mass due to the spring is Fr = -kx

Frequency of oscillationf = 1/T [s-1] = [Hz]

Period of oscillation T = 1/f [s]

Hooke’s Law

Fr = -kx

The equilibrium position is taken to be x = 0. Then the displacement is positive to the right, negative to the left. The force opposes the displacement of the spring.

Energy An oscillating mass attached to a spring has KE and PE.

The potential energy associated with a spring compressed a distance x is PE = ½ kx2

If the initial compression has x = A, then the total PE at the start is PE = ½ kA2

At any point during the oscillation, Energy = ½ kA2 = ½ kx2 + ½ mv2

Energy

PE is max at maximum displacement

PE = 0 at equilibrium position.

KE = 0 at maximum displacement

KE = max at equilibrium position

Energy

Energy

Total Energy = ½ kA2 = ½ kx2 + ½ mv2

Velocity at any position, x can be found by solving for v:

v = ±

With vmax = ±

Example

When a 0.50 kg mass is suspended vertically from a spring, the spring stretches a distance of 10 cm to a new equilibrium position. What is the spring constant of the spring? The mass is then pulled down another 5.0 cm and released. What is the highest position of the oscillating mass?

Equations of Motion

Equations of Motion

Recall ω = 2πf = 2π/T [radians/sec]

SMH ω = √(k/m) for spring mass system

SHMT = 2π√(m/k) for spring mass system

SHMT = 2π√(L/g) for simple pendulum

Equations of Motion The displacement may

depend on Sine

The displacement may depend on Cosine

Which function to choose depends on initial conditions.

Example

A mass on a spring oscillates vertically with an amplitude of 15cm, a frequency of 0.20 Hz, and an equation of motion given by y = Asinωt with

y0 = 0 and t0 = 0 and initial upward motion.

a) What are the position and direction of the mass at t = 3.1sec?

b) How many oscillations will it make in 12 seconds?

Choosing Sine or Cosine – Initial Conditions

See page 444!!

If y = 0 when t = 0 and initial displacement is upward, use y = Asin ωt

If y = A when t = 0, use y = Acos ωt If y = 0 when t = 0 and initial displacement is downward,

use y = -Asin ωt If y = -A when t = 0, use y = -Acos ωt

Position formulas for SHM

For a mass oscillating where displacement =0 when t = 0, Y = ±Asinωt

For a mass oscillating where displacement = ± A when t=0, Y = ±Acosωt

Velocity and Acceleration in SHM

Using calculus, velocity and acceleration formulas can be derived:

v = ωAcosωt (for y = Asinωt)

a = -ω2sinωt (for y = Asinωt)

Waves!

A wave is a periodic disturbance that transfers energy.

Wave Characteristics λ = wavelength [meters]

measures length crest to crest or trough to trough.

f = frequency [s-1] or [Hz]measures cycles per second

T = period [s]measures time for one complete cycle

A = amplitudemeasures displacement from equilibrium

v = velocity [m/s]measures wave speedv = f λ

Types of Waves Longitudinal Waves – wave oscillates in the same

direction as the velocity. Transverse Waves – wave oscillates at 90 degrees to the

direction of travel. Mechanical Waves – waves that travel through a medium

such as water or air (could be longitudinal or transverse) Electromagnetic Waves – waves that travel in a vacuum

(these are always transverse and result in oscillations of electric and magnetic fields)

Wave Types

Wave Properties - Interference Principle of Superposition: when waves combine,

the resulting waveform is the sum of the displacements of the individual waves at each point in the medium.

Constructive Interference – when waves add to create a larger wave.

Destructive Interference – when waves add to create a smaller wave.

Wave Properties

When a wave strikes a medium/barrier that is different, it will be reflected and/or transmitted.

If the original wave strikes a more dense or a fixed boundary, the reflected wave will be inverted.

If the original wave strikes a boundary that is less dense or movable, the reflected wave will not be inverted.

Reflection

Reflection

Refraction and Diffraction Refraction is the bending of a wave as it enters a

new medium

Diffraction is the bending of a wave around an obstacle.

Refraction and Diffraction

Standing Waves A Standing Wave occurs when an incident wave

interferes with a reflected wave and creates nodes and antinodes which appear to stand still.

Standing Waves Standing waves are generated in a rope with a driving frequency (like a drill or a hand!) The

higher the driving frequency, the more nodes.

Notice that an integer number of half-wavelengths “fit” for resonance to occur. For a rope, a node must be at each end.

Standing waves occur at natural frequencies and resonant frequencies in a particular resonator (rope, tube, etc)

The lowest natural frequency that resonates is called the fundamental frequency of the resonator.

Higher frequencies are called resonant frequencies.

The set of frequencies are called the harmonic series. (first harmonic is fundamental frequency)

Standing Waves The length, L, of the string for standing waves is an integral

number of half wavelengths:

L = n(λn/2) or λn = 2L/n where n = 1, 2, 3…

Then the natural frequencies of vibration are

fn = v/ λn = n(v/2L) = nf1

The set of frequencies, f1, f2, f3… are called harmonics of the fundamental frequency.

Standing Waves Natural frequencies of a stretched string depend on other

parameters such as tension and mass! Many musical instruments involve strings that resonate.

Wave speed on a stretched string v = √(FT/μ) where FT is tension and μ is linear density (mass/length)

Then fn = v/ λn = n(v/2L) = n/2L √(FT/μ) = nf1

Wave Speed

Wave speed, v = f λ, depends only on the medium through which the wave travels.

If f increases, λ decreases.

Resonance

When the natural frequency of an object is matched by a driving frequency, resonance occurs.

Resonance is amplification of a wave due to the matching of a natural frequency with a driving frequency.

Examples

A piano string with length of 1.15 m and mass of 20.0 g is under a tension of 6.30 X 103 N. What is the fundamental frequency of the string? What are the next two harmonics?