A uniform rod OA of length `l`, resting on smooth surface is slightly distributed from its vertical position `P` is a point on the rod whose locus is a circle during the subsequent motion of the rod, then the distance OP is equal to

Options

A. `l//2`
B. `l//3`
C. `l//4`
D. there is no such point.

Explanation

Two forces normal reaction and weight are the only forces acting on the rod during motion. Both forces are vertical. So centre of mass will fall downwards in a vertical line.
`X_(P)=x("say")=(l)/(2)costheta-dcostheta`
or `costheta=((x)/((l)/(2)-d))` .(i)
similarly
`y_(P)=y=dsintheta`
`thereforesintheta=(y)/(d)` ..(ii)
Squaring and addinng Eqs. (i) and (ii), we get
`((X^(2))/((l)/(2)-d)^(2))+(y^(2))/(d^(2))=1`
this is an equation of a circle for `d=(l)/(4)` for any other value of `d` it is equation of ellipse.