Physics: Simple Harmonic Motion - SHM

Simple Harmonic Motion(SHM)

What is Periodic motion?

The motion which repeats itself after equal intervals of time is called periodic motion. For example: the motion of planets and satellites, vibration of springs, motion of pendulum, motion of different hands in watch etc.

What is Simple Harmonic Motion?

The periodic motion in which particle moves to and fro (back and forth) such that the acceleration of the particle is directly proportional to the displacement from mean position and directed towards the mean position is called simple harmonic motion.

i.e. for SHM, a ∝ y ( where y is the displacement from mean position)

or, a = −ky ..........................(i) (Where, k is proportionality constant. Negative

sign shows that the acceleration opposes the displacement from mean position.)

SHM in terms of uniform circular motion(Characteristics of SHM):

Equation of SHM:

Consider a particle moving in a circular orbit having centre at O and radius r with uniform angular velocity ω. Let P(x, y) be the position of the particle at any time t and corresponding angular displacement is θ.

The angular velocity is given by the relation

ω = θ / t

∴ θ = ωt ........................(i)

Draw PM perpendicular to X axis and PN perpendicular to Y axis. When the particle moves in the circular orbit, its projection or foot of the perpendicular drawn on any reference line; say point N in Y- axis executes simple harmonic motion about mean position O. Similarly point M in X- axis also executes simple harmonic motion about mean position O.

From geometry,

Sinθ = y / r

or, y = r sinθ

or, y = r sinωt .......................(ii)

This is called displacement equation of SHM.

Equation (ii) is derived by assuming initial phase angle of the particle to be zero. If the initial phase angle is not zero, then the displacement equation of SHM can be expressed as,

y = r sin(ωt + φ) .......................(iii) (where φ is phase angle at t = 0)