A block X of mass 0.5 kg is held by a long massless string on a frictionless inclined plane of inclination `30^@` to the horizontal. The string is wound on a uniform solid cylindrical drum Y of mass 2kg and of radius 0.2 and of radius 0.2 m as shown in Fingure. The drum is given an initial angular velocity such that the block X starts moving up the plane.
(i) Find the tension in the string during the motion.
(ii) At a certain instant of time the magnitude of the angular velocity of Y s `10 rad s^(-1)` calculate the distance travelled by X from that instant of time until it comes to rest

Explanation

(`a`) Mass of block `m=0.5 kg`, mass of pulley `M=2 kg`, radius of pulley `R=0.2 m`

`mgsin30^(@)-T=mg` (`i`)
`TR=Ialpha=(1)/(2)MR^(2)alpha` (`ii`)
`a=Ralpha` (`iii`)
From (`ii`) and (`iii`) , ` T=(1)/(2) Maimplies a=(2T)/(M)` in (`i`)
`(mg)/(2)-T=2(m)/(M)T`
`T=(mg//2)/(1+(2m)/(m))=(0.5xx10//2)/(1+(2xx0.5)/(2))=(2.5)/(1.5)=(5)/(3)N`
When the block moves up, it is moving under retardation.
(`b`) The block will move up until the `K.E`. of (block `+` pulley) converts into the potential energy of block.

`(1)/(2)mv^(2)+(1)/(2)Iomega^(2)=mgh`
`(1)/(2)mR^(2)omega^(2)+(1)/(2)xx(1)/(2)MR^(2)omega^(2)=mg(d/2)`
`d=((M+2m)R^(2)omega^(2))/(2mg)`
`=((2+2xx0.5)(0.2)^(2)(10)^(2))/(2xx0.5xx10)`
`=1.2 m`