Simple Harmonic Motion

Simple Harmonic Motion • Vibration around an equilibrium position in which a restoring force is proportional to the displacement from equilibrium. • Back and forth motion over the same path.

Simple Harmonic Motion (cont.) • Equilibrium position – starting or resting position (x=0) • Restoring force – the force that returns the mass to the original starting position.

Hooke’s Law • Felastic = – k x F: restoring force x: displacement k: spring constant

Spring Constant • The spring constant depends on the stiffness of the spring. • Increase stiffness, increase k • negative k indicates that the direction of the restoring force is opposite the mass displacement.

Summary of SHM • At maximum displacement: • the velocity becomes zero. • the acceleration reaches maximum. • the restoring force reaches maximum. • PE is max, KE is zero.

Summary of SHM • At equilibrium: • the velocity reaches maximum. KE is max. • the acceleration becomes zero. • the restoring force becomes zero. • KE is max, PE is zero.

Potential Energy in the System • Mass-spring system • elastic PE. • Simple pendulum • gravitational PE.

Measuring Simple Harmonic Motion • Amplitude – maximum displacement from equilibrium. • Pendulum: measured by the angle between equilibrium and displacement. • spring-mass system: the amount the spring is stretched or compressed.

Measuring SHM (cont.) • Period • the time it takes to execute one complete cycle of motion. • Frequency • the number of cycles or vibrations per unit of time. • SI Unit: hertz (Hz) f = 1(or) T = 1 T f

The Period of a Simple Pendulum • T = 2  L/g L – length of string g – gravitational pull • The period is independent of the mass and the amplitude.

The Period of a Mass-Spring System • T = 2 m/k • A heavy mass will have less acceleration than a light mass increasing the time it takes to complete one cycle.

Waves • The motion of a disturbance that propagates through a medium or space. • The transfer of energy without the transfer of matter.

Characteristics of Waves • Wavelength – the distance between 2 adjacent similar points of the wave, such as from crest to crest or from trough to trough. () • Amplitude - maximum displacement on either side of the equilibrium position.

Characteristics of Waves (cont.) • Frequency – number of crests or troughs that pass a given point in a unit of time. • Period – amount of time required for one complete wavelength to pass a given point. • Wave speedv = f 

Mechanical Waves • A wave whose propagation requires the existence of a medium. • waves result from molecular movement. • energy is needed to start the disturbance. • the medium does not travel with the wave. • After the wave has passed, the particles of the medium return to original position. • Ex: sound (uses air as the medium)

Electromagnetic Wave • A wave whose propagation does not need a medium to travel through. (energy waves) • Ex: light waves

Waves (cont.) • Pulse wave – a single nonperiodic disturbance. • Periodic wave – a wave whose source is some form of periodic motion. • Sine wave – a wave that vibrates with simple harmonic motion. (named for the graph produced when graphing y = sin x).

2Types of Mechanical Waves:Transverse Waves • A wave whose particle displacement is perpendicular to the direction of the wave motion. • Ex: water waves

Transverse Waves (cont.) • crest – the highest point above the equilibrium position. • trough – the lowest point below the equilibrium position. • equilibrium line – the resting point.

Longitudinal Waves • A wave whose particles vibrate parallel to the direction of the wave motion. • Ex: sound waves • compression – the portion of the wave where particles are pushed together; particles are densely packed. • rarefaction – the portion of the wave behind the compression where the particles are stretched apart;low density of particles.

Superposition Principle • Since waves are not matter, but displacement of matter, 2 waves can occupy the same space at the same time. • When 2 or more waves travel through a medium, the resultant wave is the sum of the displacements of the individual waves at each point.

Interference • The effects produced by 2 or more waves that superpose while passing through a given region. • 2 Types of interference: • Constructive interference • Destructive interference

Constructive Interference • Interference in which individual displacements on the same side of the equilibrium position are added together to form the resultant wave. • These waves are said to be “in phase”. • When crest meets crest it increases the amplitude.

Destructive Interference • Interference in which individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave. • These waves are said to be “out of phase” • When crest meets trough, the amplitude is canceled.

Longitudinal Waves • Constructive interference • two compressions interfere to increase the net force on the particles. • Destructive interference • a compression and a rarefaction interfere reducing the net force on the particles.

Reflection • When a wave impulse reaches a boundary, the pulse travels back along the medium in the opposite direction. • Incident wave: incoming wave • Reflected wave: returning wave bounced off of a boundary

Reflected Wave Pulses • Reflected pulse from a free boundary: • The reflected pulse is identical to the incident pulse. • Reflected pulse from a fixed boundary: • The reflected pulse is inverted from the incident pulse.

Standing Waves • A wave pattern that results when 2 waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere. • These 2 waves cancel one another out at the site of destructive interference. • The crest of one wave becomes the trough of the second wave. • The result is the wave doesn’t appear to be moving.

Standing Waves (cont.) • Node – point in a standing wave that always undergoes complete destructive interference and therefore is stationary. • No matter disturbance at this point. • Antinode – point in a standing wave, halfway between 2 nodes, at which the largest amplitude occurs. (loops)

Doppler Effect • A frequency shift that is the result of relative motion between the source of waves and an observer. • When an ambulance moves toward you with its siren going: • the pitch is higher as it approaches. • the pitch is lower as it moves away.

Doppler Effect (cont.) • The pitch depends on frequency, but the frequency from the source of sound is not changing. • The perceived frequency is greater as vehicle approaches • The perceived frequency is lower as vehicle moves away.

Bow Wave • The V-shaped wave made by an object moving across a liquid surface at a speed greater than the wave speed.

Shock Wave • The cone-shaped wave made by an object moving at supersonic speed through a fluid. • Sonic boom - the loud sound resulting from the incidence of a shock wave.

Simple Harmonic MotionSample Problem 1 • If a spring is stretched 2.0 cm by a mass of 0.55 kg, calculate the force constant.

Sample Problem 2 • A 0.35 kg mass attached to a spring of spring constant 130 N/m is free to move on a frictionless horizontal surface. If the mass is released from rest at x = 0.10m, find the force on it. • Find the acceleration at x = 0.10 m

Sample Problem 3 • A spring of spring constant 30 N/m is attached to different masses and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes: 2.3 kg 15 g

Sample Problem 4 • A man needs to know the height of a tower, but darkness obscures the ceiling. He knows, however, that a long pendulum extends from the ceiling almost to the floor and that its period is 12.0s. How tall is the tower?

Sample Problem 5 • A wave traveling in the positive x direction has a wavelength of 40 cm and an amplitude of 15 cm. Find the period and speed of the wave if it has a frequency of 8.0 Hz.

Sample Problem 6 • The note middle C on a piano has a frequency of approximately 262 Hz and a wavelength in air of 1.31 m. Find the speed of sound in air.

Sample Problem 7 • An FM station broadcasts at a frequency of 100 MHz (106), with a radio wave having a wavelength of 3.00 m. Find the speed of the radio wave.

Simple Harmonic Motion