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 P be any position of the particle after t seconds and θ be the angle turned by the particle in t seconds. We know that
θ = 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'.
θ = 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
![](data:image/png;base64,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)
The velocity is maximum, when x = 0, i.e. when N passes through O (i.e. mean position).
vmax = ω.r
The acceleration of N is the component of the acceleration of P parallel to XX' and is directed
towards the center O.
![](data:image/png;base64,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)
The acceleration is maximum, when x = r, i.e. when P is at X or X'.
The velocity is maximum, when x = 0, i.e. when N passes through O (i.e. mean position).
vmax = ω.r
The acceleration of N is the component of the acceleration of P parallel to XX' and is directed
towards the center O.
The acceleration is maximum, when x = r, i.e. when P is at X or X'.
Without being wierd I would say, a diagram to go along will be much helpful.
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