Figures 14.25 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 anti-clockwise) 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) Motion is in clockwise direction and at `t=0`, particle is at P and hence initial phase `phi = +(pi)/(2)rad`
`therefore omega = (2pi)/(T)= (2pi)/(2) " "[therefore T = 2s" given "]`
`therefore omega = pi rad s^(-1)`
Equation of SHM for x-component of particle P, `x = A cos (omega t+phi)`
`therefore x= 3 cos (pi t +(pi)/(2))`
`therefore x = -3sin pi t" cm " [ therefore cos ((pi)/(2)+theta)= -sin theta]`.
(b) Here T= 4s and A= 2 m and initial phase `phi = pi rad`
`therefore omega = (2pi)/(T) = (2pi)/(4)= (pi)/(2)rad s^(-1)`
The equation of SHM for the x-component of particle P,
`x= A cos (omega t+ phi)`
`therefore x= 2 cos ((pi)/(2) t+ pi)`
`therefore x= -2 cos ((pi)/(2)t)" cm "[therefore cos (pi +theta)= -cos theta)`.