Question

A light thread with a body of mass `m` tied to its end is wound on a uniform solid cylinder of mass `M` and radius `R`. At a moment `t=0` the system is set in motion. Assuming the friction in the axle of the cylinder to be negligible, find the time dependence of
(a) the angular velocity of the cylinder and
(b) the kinetic energy of the whole system ( `M=2m`).

Answer 1

Linear motion of the block:

`mg-T=ma` (`i`)
Angular motion of the pulley:

`tau_(0)=TxxR=I_(0)alpha`
`TR=(1)/(2)xx2mR^(2)alpha=mR^(2)alpha` (`ii`)
`a=Ralpha` (`iii`)
From (`ii`) and (`iii`), `T=ma`
`mg-T=maimpliesmg-ma=maimpliesa=(g)/(2)`
`alpha=(a)/(R)=(g)/(2R)`
Angular velocity of cylinder after time `t`
`omega=omega_(0)+alpha t =0+("gt")/(2R)=("gt")/(2R)`
`K.E`. of system `= K.E`. of block `+ K.E`. of pulley
` = (1)/(2)mv^(2)+(1)/(2)I_(0)omega^(2)`
`= (1)/(2)mR^(2)omega^(2)+(1)/(2)xx(1)/(2)xx2mR^(2)omega^(2)`
`=mR^(2)omega^(2)=mR^(2)(("gt")/(2R))^(2)`
`=(1)/(4)mg^(2)t^(2)`