Classical Physics- Periodic Motion

Periodic Motion

If a moving body repeats its motion after regular intervals of time the motion is said to be harmonic or periodic. The time interval after which it repeats the motion is called time period. if the body moves to and fro on the same path, it is said to be oscillation. In simple harmonic motion the particle moves in a straight line and the acceleration of the particle is always infected towards a fixed point on the line. This fixed point is called centre of oscillation. The acceleration inSHMis given by

a = - ω²x or F = - mω²x = - Kx

K = mω²is called the force constant or spring constant.

The force which brings the particle back towards the equilibrium (or mean) position is called restoring force.

SHMmay be thought of as the projection of uniform circular motion along a diameter.

X = r cos αx

Y = r sin αx

Α = ω²x

Or d²x / dt²= - ω²x

Given the solutionx = x₀sin αx

When the particle starts from mean position,

X = x₀cos ωt

When, starts from extreme position

X = x₀sin (ωt ±Ø)

X = x₀cos (ωt ±Ø)

When it starts in between mean and extreme positions.

X = x₀e ±iωt

Wherexis called instantaneous displacement,x₀the amplitude (maximum displacement)Øinitial phase angle or epoch or angle of repose andωthe angular frequency.

Linear frequency f = ω / 2π and T = 1 / ƒ is time period

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