Vibrations and Waves

Chapter 12

Simple Harmonic Motion

• A motion that occurs repeatedly, vibrating back and forth over its equilibrium point.

• The force acting on the system must be internal and restorative, resulting in a sinusoidal wave pattern.

• There are three types of SHM that will be discussed.• Mass-Spring System• Pendulum• Waves

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Restorative Force

• The restorative force that creates SHM always points to the equilibrium point.

• Mass-Spring System • Fspring = -(spring constant)(displacement) = -kx

• Pendulum (portion of gravity that points towards equilibrium point)• Fpendulum = Fg(sin)

Describing SHM

• We can describe SHM using three measurements.

• Amplitude - amount displaced from the equilibrium point.

• Mass-spring systems and Waves - distance from E.P.

• Pendulum - degrees displaced from E.P.

Describing SHM (cont)

• Period - time it takes to complete one cycle of motion – Symbol - T– Units - seconds

• Frequency - number of cycles completed in one unit of time, usually a second. – Symbol - ƒ– Units - cycles/second or Hertz (Hz)

• Period and frequency have an inverse relationship• T = 1 / ƒ and ƒ = 1 / T

Period of a Pendulum

• The period of a pendulum is dependant on the length of the string and the acceleration due to gravity.

• T= 2π x square root of (length/acceleration due to gravity)

• T = 2π√(L/g)

• Sample Problem 12B

Period of a Mass-Spring System

• The period of a Mass-Spring System is dependent on the mass attached to the spring and the spring constant of that spring.

• T = 2π x square root of (mass/spring constant)• T = 2π√(m/k)• Sample Problem 12C

Waves

• Waves are the motion of disturbance.• As a particle in a medium is disturbed, the energy

causes nearby particles to be disturbed.• The medium provides particles that the energy can

travel through.• The particles themselves do not travel along with

the wave.• Instead, they undergo simple harmonic motion,

moving back and forth about a point of equilibrium.

Wave Types

• Pulse Wave - A single, non-periodic disturbance

• Periodic Waves - A series of waves whose source of energy is a form of simple harmonic motion

Wave Types (cont)

• Waves can also be described by the motion of the particles in relation to the motion of the wave.

• Transverse Waves - particles of medium move perpendicular to the motion of the wave– Ex. Ocean waves

• Longitudinal Waves - particles of the medium move in the same direction as the motion of the wave – Ex. Sound waves

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Describing Transverse Waves

• Crest - the high point of a transverse wave

• Trough - the lowest point of a transverse wave

• Wavelength ()- distance between two crests (or troughs) on a wave

Describing Longitudinal Waves

• Compression - point at which particles are closest together

• Rarefaction - point at which the particles are farthest apart

• Wavelength () - distance between two points of compression (or rarefaction) on a wave

Describing Waves

• Wave speed - the velocity at which a crest or trough moves through a medium.

• Period - the length of time it takes for two crests to pass by a given point.

• Frequency - the number of crests that pass through a point in a given amount of time (s)

Describing Waves (cont)

• Wave speed can be related to the frequency or period of a wave and its wavelength

• v = (frequency)(wavelength) = (ƒ)( )

• v = (wavelength)/(period) = / T

• Sample Problem 12D

Wave Interactions

• When two pulse waves encounter one another they interact.

• If the displacement of the waves is in the same direction the waves energy will add together.

• Constructive Interference• If the two waves have opposite displacements

their energy will cancel.• Destructive Interference

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Reflection

• When a wave reached the boundary of the medium it is traveling through, it is reflected.

• If the boundary is free the wave simply changes direction.

• If the boundary is fixed the wave will be inverted as it is reflected.

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Standing Waves

• When a string is fixed on both ends, a wave traveling on the string at the proper speed will engage in both constructive and destructive interference.

• This forms a pattern with areas of no movement on the string (nodes) and areas of maximum movement (antinodes)

• There can be several standing waves along a single string.

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Wavelength of Standing Waves

• The wavelength of a standing wave is the distance between three nodes.

• On a string with a single standing wave, the wavelength is 2 times the length of the string.

• The wavelength for any number of standing waves on a string can be described with an equation

= 2(length of the string)/(number of standing waves)

= 2L/(#sw)