Simple Harmonic Motion (SHM): A Quick Revision

Simple Harmonic Motion (SHM) is a specific type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium position. This results in a sinusoidal oscillation.  

Key Characteristics of SHM:

• Periodic motion: The motion repeats itself after a fixed interval of time.
• Restoring force: A force that always acts towards the equilibrium position.
• Sinusoidal motion: The displacement, velocity, and acceleration of the object vary sinusoidally with time.

Equations of SHM:

• Displacement:

  • x = A sin(ωt + φ) or x = A cos(ωt + φ)
  • A is the amplitude (maximum displacement)
  • ω is the angular frequency
  • t is time
  • φ is the phase constant

• Velocity:

  • v = ωA cos(ωt + φ) or v = -ωA sin(ωt + φ)

• Acceleration:

  • a = -ω²A sin(ωt + φ) or a = -ω²A cos(ωt + φ)

Important Terms:

• Period (T): The time taken to complete one oscillation.
• Frequency (f): The number of oscillations per unit time.
• Angular frequency (ω): 2π times the frequency.
• Amplitude (A): The maximum displacement from the equilibrium position.
• Phase constant (φ): Determines the initial position and velocity of the object.

Examples of SHM:

• A mass attached to a spring
• A simple pendulum
• The vibrations of a guitar string
• The motion of a piston in an internal combustion engine

Energy in SHM:

• Kinetic energy: Maximum at the equilibrium position and minimum at the extreme positions.
• Potential energy: Maximum at the extreme positions and minimum at the equilibrium position.
• Total mechanical energy: Remains constant in the absence of friction.