U7: Oscillatory Motion and Waves

Something that oscillates moves back and forth between two points

● Ex: a child on a swing, an ocean buoy, the beating of a heart

● All oscillations involve force and energy

● Some oscillations create waves - disturbances that move from its source and carry energy

○ Ex: earthquakes, visible light

A restoring force is a force acting in opposition to the force caused by a deformation

● Hooke’s law: 𝐹 = −𝑘𝑥

○ F is the restoring force

■ The restoring force is in the direction opposite to the displacement

○ x is the displacement from equilibrium of deformation

○ k is a force constant related to the difficulty in deforming the system

■ The larger the force constant, the greater the restoring force, and the stiffer the system

Elastic potential energy is potential energy stored as a result of deformation of an elastic object (ex: the stretching of a spring)

● 𝑃𝐸el = .5𝑘𝑥2

Periodic motion is a motion that repeats itself at regular time intervals

● The period (T) is the time to complete one oscillation (it remains constant)

● Frequency (f) is the number of events per unit time

○ 𝑓 = 1/𝑇

Simple Harmonic Motion is the name given to oscillatory motion for a system where the net force can be described by Hooke’s Law

● This system is called a simple harmonic oscillator

○ If the net force can be described by Hooke’s law, and there is no damping by friction or other non-conservative forces, then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position

● The maximum displacement from equilibrium is called the amplitude (X)

○ Period and frequency are independent of amplitude

■ 𝑇 = 2𝜋√(𝑚/𝑘)

■ 𝑓 = (1/2𝜋)√(𝑘/𝑚)

Simple harmonic motion can be related to sine and cosine curves

● 𝑥(𝑡) = 𝑋 𝑐𝑜𝑠(2𝜋𝑡)/𝑇

● 𝑣(𝑡) = −𝑣max 𝑠𝑖𝑛(2𝜋𝑡/𝑇)

● 𝑎(𝑡) = (−𝑘𝑋/𝑚)𝑐𝑜𝑠(2𝜋𝑡/𝑇)

A simple pendulum is defined to have an object that has a small mass suspended from a light wire or string

● 𝑇 = 2𝜋√𝐿/𝑔

○ The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity

○ It is independent of other factors such as mass

Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant

● .5𝑚𝑣2 + .5𝑘𝑥2 = constant

● Maximum velocity depends on three factors:

○ 𝑣𝑚𝑎𝑥 = √(𝑘/𝑚) * 𝑋

■ It is directly proportional to amplitude

■ It is greater for stiffer systems

■ It is smaller for objects that have larger masses

A transverse wave is a wave in which the disturbance is perpendicular to the direction of propagation

A longitudinal wave has a disturbance parallel to its direction of propagation

● Waves can be the combination of the two (Ex: water waves)

In waves, the fundamental frequency is the longest wavelength

● The overtones of harmonics are multiples of the fundamental frequency

Beat frequency is the frequency of the amplitude fluctuations of a wave

● Beats occur when waves of similar frequencies are superimposed

● 𝑓𝐵 = |𝑓1 − 𝑓2|