Define simple harmonic motion. Show that the motion of projection of particle performing uniform circular motion, on any diameter is simple harmonic.

Define simple harmonic motion. Show that the motion of projection of p

1.	Define simple harmonic motion. Show that the motion of projection of particle performing uniform circular motion, on any diameter is simple harmonic.
Answer» Solution :Simple harmonic motion: A body is said to be inS.H.M, If acceleration is directly proportional to its displacement. Acts opposite in direction towards a fixed point. Relation between uniform circular motion and S.H.M.:Let a particle 'P' is rotating in a circular path of radius 'r' with a uniform angular velocity `'omega'`. After time 't' it goes to a new position 'P'. Draw normals form 'P' on to the X-axis and on the Y-axis. Let ON and OM are the PROJECTIONS on X and Y axis respectively. As the paritcle is in motion it will subtend an angle `theta=omegat` at the centre. From triangle OPN ON=OP `costheta` But `OP = r and theta=omegat` `therefore` Displacement of particle. P on X-axis at any time t is `X=rcosomegat"".............(1)` From triangle OPM `OM=Y-OPsintheta` But `OP=r and theta=omegat` `therefore` Displacement of particle P on Y-axis is `Y=rsin omegat""..............(2)` As the particle rotate in a circular path the FOOT of the perpendicular OM and ON will oscillatiate with in the limits `X to X^(1) and Y to Y^(1)`. At any point the displacement of particle P is given by `OP^(2)=OM^(2)+ON^(2)` SINCE `OM=X=rcosomegat and ON=Y=r sinomegat.` So a uniform circular motion can be treated as a combination of two mutually perpendicular simple harmonic motions.