# Simple Harmonic Motion, Springs and Pendulums

## Simple Harmonic Motion, Springs and Pendulums

# Simple Harmonic Motion

- An object in
**simple harmonic motion (SHM)**repeats its motion over regular intervals of time, also known as its**period**. - SHM follows a sinusoidal pattern, it can be described using sine or cosine functions.
- In SHM, the
**restoring force**is directly proportional to the displacement of the object from its equilibrium position and acts in the direction opposite to the displacement. - The
**angular frequency (ω)**of the motion is the rate of change of the angle that the motion makes during its oscillation, measured in radians per second. - The
**amplitude**of SHM refers to the maximum displacement of the object from its equilibrium position. - The
**phase constant or phase angle (φ)**in SHM defines the starting point on the sinusoidal wave.

# Springs

- A
**spring**obeys Hooke’s Law which states that the force required to extend or compress a spring by a certain distance is proportional to that distance. - The constant of proportionality is known as the
**spring constant k**, measured in N/m. - The potential energy stored in a spring when it is either compressed or extended is given by the equation PE = 1/2 kx².
- Simple harmonic motion of a spring can be described by the equation x = A cos(ωt + φ), where x is displacement, A is the amplitude of the motion, ω is the angular frequency, and φ is the phase constant.

# Pendulums

- A
**simple pendulum**consists of a mass (the pendulum bob) attached to a string or rod of negligible mass which swings back and forth under the influence of gravity. - The time for one complete oscillation or swing back and forth, known as the
**period of a pendulum**, depends on the length of the pendulum and the acceleration due to gravity. - The period of a simple pendulum can be given by the equation T = 2π √(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
- For small displacements, a pendulum exhibits SHM.
- In real-world pendulums, damping and friction will cause the oscillations to gradually decrease in amplitude. However, in an ideal simple pendulum, the oscillations continue indefinitely.