A solid cylinder of mass M and radius .R. is mounted on a frictionless horizontal axle so that it can freely rotate about this aris. A string of negligible mass is wrapped round the cylinder and a body of mass .m.is hung from the string. The mass is released from rest. Find the tension in the string and the angular speed of cylinder as the mass falls a distance h.

A solid cylinder of mass M and radius .R. is mounted on a frictionless

1.	A solid cylinder of mass M and radius .R. is mounted on a frictionless horizontal axle so that it can freely rotate about this aris. A string of negligible mass is wrapped round the cylinder and a body of mass .m.is hung from the string. The mass is released from rest. Find the tension in the string and the angular speed of cylinder as the mass falls a distance h.
Answer» Solution :The acceleration .a. of the falling body is GIVEN by `mg-T ma ""…(1)` Torque on the cylinder is `tau=TR=I alpha` `:.T=(IALPHA)/(R )=(MR^(2))/(2)((a)/(R^(2))) "" [ :. alpha=(a)/(R )]` or `T=(Ma)/(2) ….(2)` from (1) and (2) `T=(mMg)/((M+2m))` from CONSEVATION of energy, we have `mgh=(1)/(2) mv^(2)+(1)/(2)I omega^(2)` `=(1)/(2) m (R omega)^(2)+(1)/(2) ((MR^(2))/(2)) omega^(2)` on solving `omega =[(4 mgh)/((M+2m)R^(2))]^((1)/(2))`