A wheel of rdius r and moment of inertia I about its axis is fixed at top of an inclined plane of inclination `theta` as shownin figure. A string is wrapped round the wheel and its free end supports a block of mass M which can slide on the plane. INitialy, tehwheel is rotating at a speed `omega` in direction such that the block slides pu the plane. How far wil the block move before stopping?

Explanation

Let `a` be the deceleration of the block moving up the incline. Therefore, liner deceleration of the rim of the wheel would also be a.
Angular deceleration of wheel
`alpha = (a)/(r )` ..(i)
It `T` is tension in the string, then equation of motion of block would be
`Mg sin theta - T = Ma` ...(i)
Also, `tau = T xx r = I alpha`
`:. T = (I alpha)/(r ) = (Ia)/(r^(2))` using (i)
Put in (ii)
`Mg sin theta = Ma + T = Ma + (Ia)/(r^(2))`
`= a(M+ (I)/(r^(2))) = a ((Mr^(2)+ I)/(r^(2)))`
`a = (Mg r^(2) sin theta)/(I + M r^(2))`
Initial velocity of the block up the incline
`v = r omega`
`:.` Distance moved by the block before stopping
`s = (v^(2))/(2a) = (r^(2)omega^(2)(I + Mr^(2)))/(2Mg r^(2) sin theta)`
`s = ((I + Mr^(2)) omega^(2))/(2Mg sin theta)`