Chapter 14: Vibrations and Waves Notes

Periodic motion is a motion that is repeated in a regular cycle.

Oscillatory motion is the movement of an object back and forth between two opposing points. This is a type of periodic motion because these objects follow the same path in the same amount of time. Examples: mass on a spring, a mass on a pendulum,

vibrating guitar string, and a tree swaying in the wind Simple harmonic motion happens in any system where

the force acting to restore the object to its equilibrium position is directly proportional to the displacement of the object shown. (restoring force gets bigger as the distance gets bigger)

Periodic Motion

Periodic Motion

Equilibrium Position: Where it comes to rest.

Restoring force tries to return it to equilibrium.

•Period (T) – time to repeat one complete cycle of motion•Amplitude (A) – the maximum distance the object moves from equilibrium.•Net force is zero at equilibrium and non-zero when the object is not in equilibrium.

EPE when it is stretched or compressed.Kinetic Energy (KE when it moves between the two extremes.

Hooke’s Law: F = -kx The negative sign means the force is in the

opposite direction of the stretch or compression (displacement) of the spring.

Elastic Potential Energy: EPE = ½ kx2

Gained by a spring when it is stretched or compressed.

Period of a pendulum: The period of a pendulum does NOT depend on

the mass of the bob or the amplitude.

Periodic Motion

Resonance occurs when forces are applied to a vibrating or oscillating object at equal time intervals which makes the amplitude of the wave bigger, like when a friend pushes you on a swing Resonance is used in musical instruments to

amplify sound and in clocks to increase accuracy

EX Tacoma Narrows Bridge 1940 https://www.youtube.com/watch?v=lXyG68_caV4 Start at 2:04.

Periodic Motion

EX1: A spring stretches by .18 m when a bag of potatoes weighing 56 N is suspended from its end. Determine the spring constant. How much elastic potential energy does the spring have when it is stretched this far?

F = -kx k = F/-x k = (-.56)/-(.18) = 310 N/mEPE = ½ kx² = ½ (310)(.18²) = 5 J

EX 2: A pendulum has a length of .369m. What is the period of the pendulum on top of Mount Everest (g = 9.6 m/s2).

Periodic Motion

Waves combine harmonic motion and linear motion. A wave carries energy through matter or space

without transferring matter. Waves that travel through a physical medium such

as water, air, or a rope. A wave pulse is a single bump that travels through

a medium (like shaking a rope once). However, if the disturbance continues at a constant rate, a periodic wave is generated (like shaking the rope multiple times).

Mechanical Wave Properties

Transverse Waves – a slinky moving up and down/ side to side The disturbance is perpendicular to the wave motion.  

Longitudinal wave – a slinky being pushed or pulled The disturbance is parallel to the wave motion. Ex: Sound waves

Surface wave – like the waves in the ocean The disturbance follows a circular path that is sometimes

parallel and other times perpendicular to the wave’s motion.

Mechanical Wave Properties

Amplitude (A) – maximum distance from wave’s equilibrium

Energy – stored in the wave’s amplitude; bigger amplitude = more energy transfer

Wavelength (λ) – the length of one cycle of a wave

Trough – the low point of a transverse wave Crest – the high point of a transverse wave Cycle – one repetition of motion Period (T) – the time to complete one cycle

Frequency (f) – how many cycles are in one second.

Mechanical Wave Properties 

Velocity(v) - Phase – a particular part of a

cycle In-phase: two waves are at

the same point in their cycles Out-of-phase: two waves are

at different points in their cycles

EX 3: A sound wave has a frequency of 192 Hz and travels the length of a football field, 91.4 m in 0.271 s.

What is the speed of the wave? 

What is the wavelength of the wave?

What is the period of the wave?

If the frequency were changed to 442 Hz, what would the new wavelength and period be?

Mechanical Wave Properties

The speed of a mechanical wave depends only on the medium it passes through.

Incident wave – the pulse that strikes the boundary.

Reflected wave – the wave that returns after striking a boundary.

Damping – the gradual reduction of a wave’s amplitude (the wave dying out).

Wave Behavior

Wave Behavior

Boundaries:

Soft Boundary – the wave’s energy dies out (damping)Ex: yelling into a pillow

Rigid Boundary -the

reflected wave returns

after striking a

boundary at the same

speed but is inverted.

Ex: Echos 

Changing mediums – Refraction happens when a wave bends as it entersa boundary made of a different

medium. Ex: Light entering water

Corners Diffract – Diffraction happens when a corner drags the wave slowing it down and turning it.Ex: Light

Principle of Superposition states that two waves can combine to form a new wave. The result of two waves combining is

interference.

Constructive Interference is when two waves combine to form a larger wave.

Destructive Interference happens when two waves combine to form a smaller wave.

Wave Behavior

A wave that is trapped between two boundaries that give energy to the wave is called a standing wave. Antinode – the places of greatest amplitude Node – places of no amplitude

2 antinodes equal 1 wavelength 2AN = 1λ

Wave Behavior

1 wavelength

Picturing waves in two dimensions (2D): Think of throwing a pebble into still water.

Wavefront – a line that represents the crest of a wave in two dimensions. The circles show the wavefronts. When drawn to scale, the show

the wavelength of the wave. Ray - A line drawn at a right

angle to the wavefront to show the direction the wave travels.

Wave Behavior

Law of reflection: the angle of incidence is equal to the angle of reflection.

Wave Behavior