Vibrations and Waves Chapter 13

Hooke’s Law When a spring is stretched, a restoring force is created. The restoring force is proportional to the displacement The restoring force is always directed toward the equilibrium point The restoring force is always directed away from the displacement F = -k ∙ x

Mass on a Vertical Spring If the mass hangs motionlessly… netF = ∑F = Fg + Fk = mg + kx = 0 (Fk : restoring force)

If both springs are identical, how would the displacement of A and B differ?

Simple Harmonic Motion Motion that repeats in a regular pattern over and over again The motion is driven by a net Force that obeys Hooke’s Law

Simple Harmonic Motion At equilibrium: The spring force and the mass’s acceleration become zero. The speed reaches a maximum. At maximum displacement: The spring force and the mass’s acceleration reach a maximum. The speed becomes zero.

Elastic Potential Energy Energy stored in a stretched or compressed spring PEs = ½ k x2 When you hang a mass on a spring, where does that spring PE come from?

Simple Harmonic Motion https://sites.google.com/site/physicsflash/home/shm

Simple Harmonic Motion

Simple Harmonic Motion Important Concepts Amplitude (A): the maximum displacement of the object from its equilibrium position Period (T): time it takes for the object to move through one complete cycle of motion Frequency (f): the number of complete cycles completed in a unit of time (units s-1, Hz)

Motion of a Pendulum Imagine a pendulum hanging straight down, in its neutral position If the bob is displaced at a small angle (<15°), it will swing with SHM What is the source of the restoring force? Ft = -mg sinθ

Motion of a Pendulum Describe the speed, force, acceleration as the bob crosses the equilibrium point, during SHM max speed, zero restoring force, zero acceleration Describe the speed, force, acceleration when the bob is at its maximum displacement from the equilibrium point. zero speed, max restoring force, max acceleration The period of a pendulum, at small angles, can be calculated

Motion of a Pendulum What is g? What factors effect the value of g? If there is no friction, no non-conservative forces acting on the pendulum, what can be said about it?

Motion of a Pendulum

Waves A wave is a disturbance that carries energy through matter or space. It is a transfer of energy, not matter How the energy is carried defines the type of wave.

Waves mechanical waves propagate through a medium. (e.g. sound, water waves, earthquakes) The medium is any material through which a wave travels (e.g. air, water, a spring, the Earth, or even people).

Waves In a longitudinal wave, the particles in the medium move parallel to the direction of the wave (e.g. sound waves)

Waves In a transverse wave, the particles in the medium move perpendicular to the direction of the wave (e.g. light waves)

Waves In a surface wave, the particles typically move in circular paths in the medium.

Waves You can find all three types of mechanical waves exhibited in earthquakes.

Waves Basic wave anatomy Crest Trough Amplitude Wavelength

Waves Wave speed (v) – How fast the wave is moving (the disturbance itself). Speed depends on the medium. Light: 3 x 108 m/s (300,000,000 m/s) (671,000,000 miles/hour) Sound: 343 meters/second (768 miles/hour) Fastest: solids liquids Slowest: gasses

Wave characteristics Frequency (f) – The number of waves passing by a point in a given period of time

Waves Wave equations T: period v: wave speed, velocity λ: wavelength f: frequency

Waves

Wave interactions Interference When more than one wave occupies the same space the results can vary depending upon how the waves match up

Wave interactions Constructive Interference If the troughs and crests line up (even briefly), the resulting wave will have greater amplitude

Wave interactions Constructive Interference If the troughs and crests line up (even briefly), the resulting wave will have greater amplitude

Wave interactions Destructive Interference If the crest of one wave lines up with the trough of another wave, the resulting wave will have lower amplitude.

Wave interactions Destructive Interference If the crest of one wave lines up with the trough of another wave, the resulting wave will have lower amplitude.

Waves Waves on strings The speed of a wave on a string is affected by tension force and the composition of the string F: tension in the string μ: linear density (mass per length)

Waves Waves on Strings When a wave encounters an object it can be reflected Reflected waves in strings bounce off the surface of a rigid obstacle and are inverted, if the end of the string is fixed

Waves Waves on Strings When a wave encounters an object it can be reflected if the end of the string is free to move at the boundary, reflected waves in strings bounce off the surface of a rigid obstacle and are not inverted