Figures,. Correspond to two circular motions. The radius of the circle, the peirod of revolution, the initial position, and the sense of revolution (i.e. clockwise or anticlockwise) are indicated on each. Obtain the corresponding equations of simple harmonic motions of the revolving particle P in each case.

Explanation

(1) Let A be any point on the circle of reference of the fig. (a) From A, draw BN perpen-dicular on x-axis.
if `" "anglePOA=theta`,then
`angleOAM = theta=omegat`
`therefore` In triangle OAM,
`(OM)/(OA) = sin theta`
`therefore " "(-x)/(3)=sin omegat=sin.(2pi)/(T) t`
`therefore " "x = - 3 sin .(2pi)/(2)t" or "x = - 3`
`sin pit`
which is the equation of SHM.
(2) Let B be any point on the circle of reference of fig. (b). From B, draw BN perpendicular on x-axis.
Then `angleBON = theta=omegat`
`therefore ln DeltaONB, cos theta = (ON)/(OB)`
`therefore - x = 2 cos omega t`
`implies x = - 2 cos (2pi)/(T)t=-2cos (2pi)/(4)t`
`therefore x = - 2 cos (pi)/(4) t` which is equation of SHM