Consider the arrangement shown in figure. The string is wrapped around a uniform cylinder which rolls without slipping. The other end of the string is passed over a masslessm frictionless pulley to a falling weight, determine the acceleration of the falling mass `m` in terms of only the mass of the cylinder `M`, the mass `m` and `g`

Explanation

Let T be the tension the string and f the force of (static) friction, between the cylinder and the surface ` a_(1) =` acceleration of centre of mas of cylinder towards right
`a_(2) = ` downward acceleration of block m
`alpha = ` angular acceleration of cylinder (clockwise)
Equations of motion are :
For block mg - T ` = ma_(2) " " . . . (i)`
For cylinder, T + f ` = ma_(1) " " . . . (ii)`
`alpha = ((T - f)R)/((1)/(2) MR^(2))" ". . . (iii) `
The string attaches the mass m to the highest point of the cylinder, hence
`v_(m) = v_("COM") + Romega`
Differentiating, we get
`a_(2) = a_(1) + R alpha " ". . . (iv)`
We also have (for rolling without slipping)
`a_(1) = Ralpha " " . . . (v) `
Solving these equations, we get `a_(2) = (8mg)/(3M + 8m)`