A metallic plate has been fabricated by welding two semicircular discs - `D_(1)` and `D_(2)` of radii R and `(R)/(2)` respectively (fig. a). O and O' are the centre of curvature of the two discs and each disc has a mass 6m. The plate is in `xy` plane. Now the plate is folded along the y axis so as to bring the part OAB in `yz` plane. (fig. b). The plate is now set free to be able to rotate freely about the z – axis. A particle of mass `m`, moving with a velocity `v` in the `xy` plane along the line `x = d` hits the plate and sticks to it `(d lt R)` . Just before collision speed of the particle was v.
(a)
(a) Find the moment of inertia of inertia of the plate about z axis.
(b) Find the angular speed of the plate after collision.

Answer

(a) `(9)/(2) mR^(2)`
(b) `omega = (2vd)/(11R^(2))`