Two metal discs, one with radius `r` and mass `m` and the other with radius `2r` and mass `2m`, are welded together and mounted on a frictionless axis through their common center.
(a) What is the total moment of inertia of the two discs?
(b) A light string is wrapped around the edge of the smaller disc and a block of mass `m_(0)` is suspended from the free end of the string. If the block is released from rest at a distance `h`, what is its speed just before it strikes the floor?

Explanation

(`a`) `I_(1)=(1)/(2)mr^(2), I_(2)=(1)/(2)xx2mxx(2r)^(2)=4mr^(2)`
`I=I_(1)+I_(2)=(9mr^(2))/(2)`
(`b`) Loss in the gravitational potential of block `=` gain in the kinetic energy of (block `+` disc)
`m_(0)gh=(1)/(2)m_(0)v^(2)+(1)/(2)Iomega^(2)`
`v=romegaimplies omega= (v)/(r)`
`=(1)/(2)m_(0)v^(2)+(1)/(2)xx(9mr^(2))/(2)xx(v^(2))/(r^(2))`
`((1)/(2)m_(0)+(9)/(4)m)v^(2)=((2m_(0)+9m)/(4))v^(2)`
`v=sqrt((4m_(0)gh)/(9m+2m_(0)))`