Chapter 12
Section One (Simple Harmonic Motion):

Hooke's law: states that for a given spring the force the spring exerts is proportional to the negative of the displacement that the spring is stretched from rest
Simple harmonic motion: the repetitive, back-and-forth motion of an object such as a pendulum or a mass oscillating on the end of a spring.
Restoration force: force that pushes the object back toward the equilibrium position. It causes the motion to pass through the equilibrium position twice each cycle.

F = -kx

The displacement (x), velocity (v), and acceleration (a) are all sinusoidal graphs.

Section Two (Measuring Simple Harmonic Motion):

Amplitude: harmonic motion measures how far the object moves
Period (T): the time that it takes for one complete cycle of motion
Frequency (f): the number of complete cycles of motion occurring in one second. Measured in Hertz (Hz).

For a simple pendulum, the period of small oscillations is given by:

For a mass-spring system with mass (m) and spring constant (k), the period of oscillation is:

Section Three (Properties of Waves):


Mechanical waves: require the presence of a medium to pass through
*An example of a medium is a ripple traveling on the surface of a pond, or a pulse traveling along a stretched spring
Transverse wave: the vibrations of a wave move perpendicular to the direction of motion of the wave
Longitudinal wave: vibrations are parallel to the direction of motion
Wavelength: the shortest distance between corresponding parts of two waves
Trough: the lowest point of a sinusoidal wave
Crest: the highest point of a sinusoidal wave
Wave Speed: the speed with which a wave propagates
Amplitude: the energy transferred by a wave is proportional to the square of this

Section 4 (Wave Interactions):

The Law of Superposition: The total displacement of the medium is equal to the sum of the displacements of the overlapping waves at each point
Constructive interference: When two waves overlap with displacements in the same direction, the resulting wave has an amplitude greater than either of the two overlapping waves
Destructive interference: When two overlapping waves have displacements in opposite directions the resulting wave will have a displacement less than the displacement of the wave with larger amplitude, or no displacement at all.
Standing wave: when a wave is traveling in a confined space with just the right frequency. This results from a wave constructively and destructively interfering with its own reflection
Nodes: the end points of a wave where complete destructive interference occurs
Antinode: placed between every pair of nodes where constructive interference causes he osicillation to reach a relative maximum amplitude

When a wave...
reaches a fixed boundary...the wave will reflect back inverted with respect to the initial wave
reflects off a free boundary... the wave will reflect off the boundary in the same orientation that it arrived

When trying to solve for the length or wavelength of a wave, use these principles:
When there is one node, wavelength= 2L
When there are two nodes, wavelength= L
When there are three nodes, wavelength= 2/3L

*the nth harmonic will have n + 1 nodes and n antinodes

Fun Practice Problems that you SHOULD TRY:
(Section 1)
1. A slingshot consists of a light leather cup attached between two rubber bands. If it takes a force of 32 Newtons to stretch the bands 1.2 cm, what is the equivalent spring constant of the rubber bands?
2. A pinball machine uses a spring that is compressed 4.0 cm to launch a ball. If the spring constant is 13 N/m, what is the force on the ball at the moment the spring is released?

(Section 2)
1. You are designing a pendulum clock to have a period of 1.0 s, how long should the pendulum be?
2. A 125 N object vibrates with a period of 3.56 s when hanging from a spring, what is the spring constant of the spring?

(Section 3)
1. A piano emits frequencies that range from a low of about 28 Hz to a high of about 4200 Hz, find the range of wavelengths in air obtained by this instrument when the speed of sound in air is 340 m/s.
2. The smallest insects that bat can detect are approximately the size of one wavelength of sound the bat makes. What is the minimum frequency of sound waves required for the bat to detect an insect that is .57 cm long?

(Section 4)
1. A wave of amplitude 0.30 m interferes with a second wave of amplitude 0.20 m. What is the largest resultant displacement that may occur?

Faughn, Jerry S., and Raymond A. Serway. Physics. Austin: Harcourt Classroom Education Company, 2002.