A uniform disc of radius a and mass m is rotating freely with an angular speed of in a horizontal plane about a smooth fixed vertical axis passing through its centre. A particle, also of mass m is suddenly attached to the rim of the disc and starts to rotate with it. The new angular speed of this system is

Step-by-step solution

Given, mass of disc =m Radius of disc =a Angular speed = Moment of inertia of the disc, I= 1 2 m a^2 Initial angular momentum, L_i=I = 1 2 m a^2 Let ^ be the angular momentum of the disc when the particle of mass m is attached at the rim. Final angular momentum, L_f=I ^ +m a v^ aligned & = 1 2 m a^2 ^ +m a^2 ^ & [ v^ = ^ a ] aligned Applying conservation of angular momentum, aligned L_i & =L_f 1 2 m a^2 & = 1 2 m a^2 ^ +m a^2 ^ ^ & = 3 aligned