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 xF: restoring force

x: displacementk: 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/gL – 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 speed v = 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 kg15 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.