A cylinder of mass m is suspended through two strings wrapped around it as shwon in figure . Find (a). the tension T in the string and (b). the speed o the cylinder as it falls through a distance h.

Explanation

The portion of the strings between ceiling and cylinder are at rest. Hence the points of the cylinder where the strings leave it are at rest also. The cylinder is thus rolling without slipping on the strings. Suppose the centre of cylinder falls with an acceleration a. The angular acceleration of cylinder about its axis given by
`alpha = (a)/(R) " " . . .(i)`
as the cylinder does not slip over the strings.
The equation of motion for the centre of mass of cylinder is
Mg - 2T = Ma
and for the motion about the centre of mass it is

` 2 T . R = ((MR^(2))/(2)) alpha, "where" I = (MR^(2))/(2)`
`2 T R = (MR^(2))/(2) (a)/(R) rArr 2 T = (Ma)/(2) " ". . . (ii)`
From (i) and (ii) on adding
`Mg = (Ma)/(2) + Ma , (3a)/(2) = g `
`a = (2g)/(3)`
`:. 2 T = (M)/(2) (2g)/(3) rArr T = (Mg)/(6) `
As the centre of cylinder starts moving from rest, the velocity after it has fallen a height h is given by

`v^(2) = 2 [(2g)/(3)] h "or" v = sqrt((4gh)/(3))`