A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.

Explanation

The moment of inertia of the hollow cylinder about it axis
`I=MR^(2)` [`because` can considered as ring]
`3xx(0.4)^(2)`
`=0.48kgm^(2)`
Torque on hollow cylinder
`tau=RFsin90^(@)`
[`because` Rope is as tangent to cylinder, `theta=90^(@)`]
`=RF [because sin90^(@)=1]`
`=0.4xx30`
`=12NM`
Now `tau=Ialphaimpliesalpha=(tau)/(I)`
`therefore alpha=(12)/(0.48)=25" rad s"^(-2)` and tangential acceleration `a_(1)=Ralpha` and radial acceleration `a_(r )=0`
`therefore a=sqrt(a_(T)^(2)+a_(r)^(2))`
= `sqrt(((Ralpha)^(2)+(0)^(2))^(2))`
`=sqrt((0.4xx25)^(2))`
`=sqrt((10)^(2))`
`therefore a=10ms^(-2)`