A particle is performing simple harmonic motion of period T about a point O and it passes through a point P, where OP=y with velicity v in the direction vecOP. Show that the time which elapses before it returns to P again is (T//pi)tan^(-1)((vT)/(2piy))

A particle is performing simple harmonic motion of period T about a po

1.	A particle is performing simple harmonic motion of period T about a point O and it passes through a point P, where OP=y with velicity v in the direction vecOP. Show that the time which elapses before it returns to P again is (T//pi)tan^(-1)((vT)/(2piy))
Answer» Solution : let Q and R be the positions of particle of REFERENCE on the circleof reference, corresponding to vibrating point at P in SHM, in the FIRST instant and when it returns to p again. Here, `y=rsin omega t` VELOCITY, `v=(dy)/(dt)romegacosomegat` `:. (y)/(v)=(1)/(omega)tan omega t or tan omegat =(omegay)/(v)` or `omegat=tan^(1)((omegay)/(v))` Required time, `t=(/_QOP)/(omega)=(2(90^(@)-omegat))/(omega)` `=(2[90^(@)-tan^(-1)((omegay)/(v))])/(omega)=(2TAN^(-1)((v)/(omegay)))/(omega)` `=(2tan^(-1)((v)/(y)xx(T)/(2pi)))/(2pi//T)=(T)/(pi)tan^(-1)((VT)/(2piy))proved`