Mechanics 6

Waves and Oscillations: Learning Goals

• Topics include Hooke’s Law, simple harmonic motion, wave motion, superposition, and interference.
• Simple harmonic motion covers energy, period, frequency, springs, and the simple pendulum.
• Wave motion covers relations to simple harmonic motion, transverse and longitudinal waves.

Hooke's Law

• Length change is proportional to the restoring force in springs and other materials.
• The equation for Hooke's Law is F = kx, where F is force, k is the spring constant, and x is the length change.
• k is the spring constant.

Hooke's Law: Examples

• If k = 300 N/m, the restoring force can be calculated for given x values (e.g., +0.1 m, +0.2 m, -0.2 m).
• The spring constant can be determined, when given a bone that bends 4 mm, when a 2 kg mass is suspended from it.

Energy in Hooke's Law Deformations

• As a spring is stretched or compressed by a force, F, the work done on the spring by the force is W = F * x.
• W = (1/2) * k * x^2
• The potential energy (PE) stored in the spring can be expressed as PE = (1/2) * k * x^2.

Energy in Hooke's Law Deformations: Example

• A toy gun's spring with k = 50 N/m is compressed by 0.15 m to fire a 2 g plastic bullet.
• PE = (1/2) * (50 N/m) * (0.15 m)^2 = 0.563 J.
• Potential energy converts to kinetic energy (KE).
• KE = (1/2) * m * v^2 = 0.563 J
• v = 23.72 m/s = 24 m/s, solving for v to find the bullet's speed as it leaves the gun.

Simple Harmonic Motion

• A mass attached to a spring experiences a restoring force towards its equilibrium position if displaced, causing it to oscillate.
• Described as sinusoidal with simple harmonic motion, if F = -kx (very common)
• The time for each full cycle of oscillation is the period, T in seconds.
• Frequency, f, is the number of cycles per second.
• f = 1/T
• T = 1/f

Energy Conservation

• In the general position: Total Energy = PE + KE
• The total energy in the (general) position = (1/2) kx² + (1/2) mv².
• At ends (x = ± A): E = (1/2) kA².
• At the center (x = 0): E = (1/2) mvmax².
• Total energy equation: E = (1/2) kA² = (1/2) mvmax² = (1/2) kx² + (1/2) mv².

Period and Frequency of SHM

• vmax = √(k/m) * A
• Period: T = 2π√(m/k)
• Frequence: f= (1/2π) * √(k/m)

The (simple) Pendulum

• Restoring force: TH = -kx, where k = mg/L.
• T = 2π√(L/g)
• Simple pendulum

The (simple) Pendulum: Example

• If T = 2π√(L/g), then T=1 = 2π√(L/(9.8))
• L = (9.8) * (1/2π)^2 = 0.25m

SHM and Wave Motion

• Sign conventions indicate positive downwards and negative upwards.

Waves: Frequency, Wavelength, Velocity

• Crests at P and Q move steadily to the right.
• Crest moves distance, λ, from P to Q in one period, T.
• Wave velocity: vwave = λ/T = fλ, since 1/T = f
• Generally, v = fλ for any wave

Transverse Waves

• The oscillation is transverse (perpendicular) to the propagation direction.

Longitudinal Waves

• Oscillation is in direction of propagation.

Superposition of Waves

• Pure constructive interference occurs when waves 1 & 2 are in-phase.
• Pure destructive interference occurs when waves 1 & 2 are out-of-phase.

Superposition & Interference of Waves

• Two sources, S₁ and S₂, are in phase.
• D₂ - D₁ = ΔD = mλ, where m = 0, 1, 2
• There is constructive interference at P

Superposition & Interference of Waves

• Two sources, S₁ and S₂, are in phase.
• ΔD = D₂ - D₁ = (m + 0.5)λ, where m = 0, 1, 2…
• There is destructive interference at P

Reflection and Standing Waves

• A string is fixed to the wall at its far end with a wave reflecting with a 180° phase change.

Standing Waves

• Incoming and outgoing waves with a string (standing wave)
• Nodes are places where the string is not moving.
• Antinodes are places where the string is moving.

Standing Waves

• Nodes = string not moving
• Antinodes = string moving

Beats

• Destructive interference and contructive interference.

Energy in Waves

• Energy and power increase as the square of amplitude: PE = (1/2) kA².