Question

show that simple harmonic motion may be regarded as the projection of uniform circular motion along a diameter of the circle.hence derive an expression for the velocity of a particle in shm?

Answer 1

In simple harmonic motion, the equation of motion is

  d²x/dt² = - ω² x ,
    ω= angular frequency, x = displacement from the mean position, t = time 

Solution is :   x = A Cos ωt        for x=A, t=0 initial position.

When a particle is moving along a circular path of radius r, the position of the projection along the diameter (x axis) is the x coordinate. The speed v is constant. So the angular speed = ω = v/r. Angle made by the radial vector with x axis = Ф.  As ω is constant, Ф = ωt
     x = r cosФ = r cos Фt

comparing the equations of motion of SHM and this equation, we say that the projection does a SHM with  amplitude = r = radius.

    v = dx/dt = - Aω sin ωt = -ω √(A²-x²)

Answer 2

The answer is attached and give like and leave a comment