Simple Harmonic Motion and Wave Properties

Vocabulary flashcards covering simple harmonic motion (SHM), mechanical and electromagnetic waves, wave properties, Hooke's Law, energy conservation, and damped oscillations.

Simple Harmonic Motion (SHM)

Periodic motion that can be modeled with a sine or cosine wave and involves a restoring force.

Mechanical Waves

Waves that travel through a medium, such as air, water, or stone.

Electromagnetic Waves

Light waves which travel through a medium (matter) or in a vacuum, such as visible light and x-rays.

Transverse Wave

A wave where the motion of matter is perpendicular to the motion of the wave (for example, a slinky coil moving up and down while the wave travels to the right).

Longitudinal Wave

A wave where the motion of matter is parallel to the motion of the wave (for example, slinky coils moving side to side while the wave travels to the right).

Amplitude (Wave Property)

The maximum displacement from the equilibrium position, measured in meters.

Wavelength ($\lambda$)

The horizontal length of $1$ cycle, measured in meters.

Period ($T$)

The time required for $1$ complete cycle, measured in seconds.

Frequency ($f$)

The number of cycles or oscillations per second, measured in units of $1/\text{sec}$ or Hertz ($\text{Hz}$), where $f = \frac{1}{T}$.

Displacement ($x$)

For a mass on a spring, the distance away from the equilibrium position (sometimes called the rest position).

Equilibrium Position ($x = 0$)

The midpoint of the range of motion; for a horizontal mass on a spring, it is the position of the free end when not compressed or stretched, while for a vertical mass, it is when stretched under gravity.

Restoring Force

A force that points towards and acts to return an oscillating system back to its equilibrium position.

Hooke's Law

The principle stating that when a spring is stretched or compressed, the spring restoring force is linearly proportional to the magnitude of displacement ($x$) but opposite in direction: $F_{\text{restoring}} = -kx$.

Spring Constant ($k$)

A measure of the stiffness of a spring, having units of $\text{N/m}$.

Restoring Force in Different Systems

For a spring, it is the spring force ($-kx$); for a swinging pendulum, it is gravity; and for sound waves, it is the elasticity of the material.

Amplitude in SHM

Half of the total range of motion, where the object's displacement varies between $x = -A$ and $x = +A$.

Frequency of a Spring in SHM

Defined by the equation $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$; frequency is proportional to $\sqrt{k}$ when mass is constant, and inversely proportional to $\sqrt{m}$ when stiffness is constant.

Energy Distribution in SHM

At maximum stretch, elastic energy is at a maximum and kinetic energy is $0$; at the equilibrium position, elastic energy is at a minimum and kinetic energy is at a maximum.

Conservation of Mechanical Energy ($E_{\text{mech}}$) in SHM

Occurs when there are no resistive forces: $\frac{1}{2}mv^2 + \frac{1}{2}kx^2 = E_{\text{mech}} = \text{constant}$, and when fully stretched, $E_{\text{mech}} = \frac{1}{2}kA^2$.

Forced (Driven) Oscillation

Motion occurring when resistive forces are present, but outside forces do work to replenish the energy being removed.

Damped Oscillations

Oscillatory motion where mechanical energy ($E_{\text{mech}}$) is not conserved due to resistive forces, causing energy and amplitude to decrease with time until reaching $0$.