Simple Harmonic Motion Basics

What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion (SHM) is a type of periodic motion where an object moves back and forth over the same path. It is characterized by its sinusoidal oscillation, meaning it moves in a smooth, wave-like pattern.

Some examples of SHM include:

• A pendulum swinging back and forth.
• A mass attached to a spring oscillating up and down.
• Vibrations of a tuning fork.

Key Characteristics of SHM

• Restoring Force: The force that brings the object back to its equilibrium position is directly proportional to the displacement and acts in the opposite direction. This is given by Hooke's Law: $F = -kx$, where $k$ is the spring constant and $x$ is the displacement.
• Equilibrium Position: The central point around which the object oscillates.
• Amplitude: The maximum displacement from the equilibrium position.
• Period $(T)$: The time taken to complete one full cycle of motion.
• Frequency $(f)$: The number of cycles per second, given by $f = \frac{1}{T}$.

Mathematical Representation of SHM

The motion of an object in SHM can be described using the following equations:

• Displacement: $x(t) = A \cos(\omega t + \phi)$
• Velocity: $v(t) = -A\omega \sin(\omega t + \phi)$
• Acceleration: $a(t) = -A\omega^2 \cos(\omega t + \phi)$

Where:

• $A$ is the amplitude.
• $\omega$ is the angular frequency, calculated as $\omega = 2\pi f$.
• $\phi$ is the phase angle, determining the starting position of the motion.