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.

Answer

C

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.