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Simple Harmonic Motion

SHM is a kind of periodic motion. We can say that all SHM's are periodic but all periodic motion are not SHM. 

Conditions for SHM:

  • The acceleration of the body or the particle should be proportional to the negative of displacement.
  • The total Mechanical Energy of the system remains conserved.
  • There must be a state of stable equilibrium.

Equation of SHM

Consider a particle revolving around a circular path of radius ''a''. with a constant angular velocity ω.
Let at time t=0,

the particle is at the position A then the projection of the particle will be at the point B on the y-axis. So when the particle comes at the top the projection also comes at the top.
Hence we can see that the particle is doing periodic motion but the projection is doing SHM. Hence the time period of the SHM is same as the time period of the circular particle.

The derivation of the actual equation of the SHM is a bit complex and it is difficult for me to show it here so for that you can refer any renowned book like HC Verma for help.

Y=Asin(ωt+Φ)

where Y show the displacement of the particle from its mean position.
A is the amplitude or maximum displacement of the particle from its mean position.
(ωt+Φ) shows the actual phase of the particle and the Φ here shows the initial phase of the particle.

Time period of some important SHM's 

Related image
  • Spring-mass system
The spring-mass system also performs SHM until no external force is applied to it.
Time Period(T)=2π √(m/k)



  • Bob and Pendulum
The simple pendulum also performs SHM
but the angular amplitude should be very small.
T=2π √(l/g)

I hope this gives you a nice idea about SHM. I will soon post a new post on Waves. 

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