SHM as Projection of Circular Motion

SHM as a projection of circular motion :

   

Let a particle be moving uniformly on a circle of radius A with angular speed ω.

If at t = 0, particle starts its motion from x-axis and in time t , it goes to p , then we have

x = OQ = A Cosθ = A Cosωt

y = OR = A Sinθ = A Sinωt

where Q and R feet of the perpendiculars drawn from P on diameter along the X-axis and the Y-axis respectively

From above equation it can be said that Q and R performing SHM about O along the x-axis and the Y-axis respectively with the same angular speed ω.

From figure (b) centripetal acceleration of the particle at P is Aω2.

Resolving this acceleration along PR and PQ

aR = ω2Asinωt = ω2x

aQ = ω2Acosωt = ω2y

The direction of aR is opposite to X and the direction aQ is opposite to Y.

Therefore, also from acceleration point of view, it can be said that Q and R are performing SHM.

Also Read :

→ Stable , Unstable & Neutral Equilibrium
→ S.H.M :Linear SHM & Angular SHM
→ Analytical Treatment to SHM
→ Kinetic Energy & Potential Energy & Total Energy in SHM
→ Average Value of P.E. & K.E. of Harmonic Oscillator
→ Simple Pendulum in Inertial & Non Inertial Frame
→ Time period of a Long Pendulum
→ SHM of Spring Mass System
→ Physical Pendulum & Torsional Pendulum
→ Undamped & Damped simple harmonic oscillations

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