Figure below correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anticlockwise) are indicated on each figure. Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P in each case.

Explanation

(a) At t = 0, OP makes and angle of `(pi)/(2)` with the X-axis. Therefore,
`ph = pi//2 `
in clockwise direction
The x projection of OP at time t gives the equation of S.H.M. as
`x = a cos (omegat + phi)`
` = a cos ((2pi)/(T)t + (pi)/(2))`
` = -a sin (2pi)/(T)t`
`a = 3 cm, T = 2 s`
`x = -3 sin pit`
In figure (b), T = 4, s, a = 2 m
At t = 0, OP makes an angle `pi` with positive direction of x-axis.
Therefore, x projection of OP at time t will give S.H.M. as
`x = a cos ((2pit)/(T) + phi)`
` = 2 cos ((2pit)/(4) + pi)`
` = -2 cos ((pi)/(2)t)`