Figures depicts two circular motions. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated on 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 = 0, OP makes an angle of `45^(@) = pi//4` rad with the (positive direction of ) x-axis. After time t, it covers an angle `(2pi)/Tt` in the anticlockwise sense, and makes an angle of `(2pi)/Tt+pi/4` with the x-axis.
The projection of OP on the x-axis at time t is given by,
`x(t)=Acos((2pi)/Tt+pi/4)`
For `T=4s`.
`x(t)=Acos((2pi)/4t+pi/4)`
which is a SHM of amplitude A, period 4 s,
and an initial phase*=`pi/4`.
(b) In this case at t = 0, OP makes an angle of `90^(@)=pi/2` with the x-axis. After a time t, it covers an angle of `(2pi)/Tt` in the clockwise sense and makes an angle of `(pi/2-(2pi)/Tt)` with the x-axis. The projection of OP on the x-axis at time t is given by
`x(t)=Bcos(pi/2-(2pi)/Tt)`
`=Bsin((2pi)/Tt)`
For T = 30 s,
`x(t)=Bsin(pi/15t)`
Writing this as x (t ) = B cos`(pi/15t-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`.