Fig.(a) and (b) depict two circular motions. The radius of the circle, the period of revolution, the initial position and sense of revolution are indicated in the figures. Obtain the simple harmonic motions of the x-projection of the radius vector of the rotating particle P in each case.

Explanation

(a) At `t=O, OP` makes an angle of `45^(@) = pi//4` rad with the (positive direction of) x-axis. After time t, if covers an angle `(2pi)/(T)t` in the anticlockise sense, and makes an angle of `(2pi)/(T)t+(pi)/(4)` with the x-axis
The projection of OP on the x-axis at time t is given by,
`x(t) = A cos ((2pi)/(T)t+(pi)/(4))`
For `T=4s`
`x(t)=A cos ((2pi)/(4)t+(pi)/(4))`
which is a SHM of amplitude A, period 4 s, and an initial `"phase"^(**) = (pi)/(4)`.
(b) In this case at t = O, OP makes an angle of `90^(@)=(pi)/(2)` with th x-axis. After a time t, it covers an angle of `(2pi)/(T)t` in the clockwise sense and makes an angle of `((pi)/(2)-(2pi)/(T)t)` with the x-axis. The projection of OP on the x-axis at time t is given by
`x (t)=B cos ((pi)/(2)-(2pi)/(T)t)`
`= B sin ((2pi)/(T)t)`
For `T = 30 s`
`x(t)=B sin ((pi)/(15)t)`
Writing this as `x(t) = B cos ((pi)/(15)t-(pi)/(2))`, and comparing with Eq. (14.4) We find that this represents a SHM of amplitude B, period 30 s, and an initial phase of `-(pi)/(2)`.