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.