Figure 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 figure. Obtain the SHMs of the x-projection of the radius vector of the rotating particle P in each case.

Explanation

(a) At `t=0` , OP makes anangleof `30^(@) = (pi)/( 6)` rad with the positive direction of x-axid. After time t, it covers an angle `( 2pi)/(T)t` in anticlockwise direction, and makes an angle of `(2pi)/(T)t+(pi)/(6) ` 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)/(6))`
For`t=2s`
`x(t) = Acos((2pi)/(2)t + (pi)/(6))`
Which is an SHM of amplitude A, period 2s, and an initial phase `= (pi)/(6)`
(b) At `t=0`, OP makes an angle of `90^(@) = (pi)/(2)` with the x-axis . After time t, it covers anangle of `(2pi)/(T)t`, in the clockwise direction and makes an angle of `((pi)/(2) - (2pi)/(T)t)` with the x-axis . The projection of OP on x-axis at time t, is given by
`x(t) = B cos ((pi)/(2) - (2pi)/(T) t)`
`= B sin ((2pi)/(T)t)`
For T= 1 min = 60s
`x(t) = B sin((pi)/(30)t)`
Writing this as `x(t) = Bcos ((pi)/(30t)-(pi)/(2))` ( as ` cos ( - theta) = cos theta` and `cos ( 90^(@)- theta) = sintheta )` and comparing it with equation (ii),`x(t) = A cos ( omega t + phi t)`, we find that this represents SHM of amplitude B, period 60s , and initial phase of `(-(pi)/(2))`