SHM

Definitions

Period, T: Time taken for one complete oscillation. Measured in seconds. 

Frequency, f: Number of complete oscillations per unit time. Measured in Hertz (s^-1).

Amplitude, A: The maximum displacement from the equilibrium position. Measured in metres.

Angular Frequency, ω: Angular displacement per unit time. Measured in rad s^-1.

SHM

A particle is said to be executing SHM if:

• it's moving so that its acceleration is always directed towards a fixed point
• and that the acceleration is directly proportional to its displacement in the opposite direction. 
• a = -ω^2x

Equations for SHM

• a = -ω^2x (defining equation)
• x = Acos(ωt + ε)
• v = -ωAsin(ωt + ε)
• a = -ω^2Acos(ωt + ε)
• vmax = +-ωA
• amax = +-ω^2A
• ω = 2π/T OR ω = 2πf 
  • Although this equation is identical to the one in circular motion, it should be noted that angular velocity (in circular motion) and angular freuency (in SHM) have a different physical signficance. 

Graphical Representation of SHM

Since velocity is the rate of change of displacement, the gradient of a distance-time graph at any instant will give the velocity at that instant. 

• Velocity is at its maximum at the equilibrium position and zero at the amplitudes

Since acceleration is the rate of change of velocity, the gradient of a velocity-time graph at any instant will give the acceleration at that instant. 

• Acceleration is at its maximum at the amplitudes and zero at the equilibrium position. 

Examples of SHM

Mass Attached to a Horizontal Helical Spring

Suppose a mass is pulled to the right and released. It will oscillate about its equilibrium position as the string is stretched and compressed. At an extension, x, there will be a tension in the spring. Assuming the spring obeys Hooke's law, the tension is given by T…