### Velocity and Acceleration of a Particle Moving With Simple Harmonic Motion

For Velocity and Acceleration of a Particle Moving With Simple Harmonic Motion (SHM), consider a particle, moving round the circumference of a circle of radius r, with a uniform angular velocity ω rad/s, as shown in Fig. 1.32. Let

θ = ω.t, and x = rcosθ = rcosωt

The velocity of N (which is the projection of

*P*be any position of the particle after t seconds and θ be the angle turned by the particle in t seconds. We know thatθ = ω.t, and x = rcosθ = rcosωt

The velocity of N (which is the projection of

*P*on*XX'*) is the component of the velocity of*P*parallel to*XX'*.
VN = v sinθ = ω.rsinθ

The velocity is maximum, when x = 0, i.e. when

v

The acceleration of

towards the center O.

The acceleration is maximum, when x = r, i.e. when P is at

The velocity is maximum, when x = 0, i.e. when

*N*passes through O (i.e. mean position).v

*max*= ω.rThe acceleration of

*N*is the component of the acceleration of P parallel to*XX'*and is directedtowards the center O.

The acceleration is maximum, when x = r, i.e. when P is at

*X or X'*.
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