A carpet of mass 'm' made of inextensible material is rolled along its length in the form of a cylinder of radius 'R' and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a very small push is given to it. Calculate the horizontal velocity of the axis of cylindrical part of the carpet when its radius reduces to R/2 .

A carpet of mass 'm' made of inextensible material is rolled along its

1.	A carpet of mass 'm' made of inextensible material is rolled along its length in the form of a cylinder of radius 'R' and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a very small push is given to it. Calculate the horizontal velocity of the axis of cylindrical part of the carpet when its radius reduces to R/2 .
Answer» Solution :Lose in PE DUE to unrolling when radius changes from `R` to `(R )/(2)` is `=MgR-((MgR)/(8))=(7)/(8)(MgR)` The lose in PE is equal to the gain in KE which is `K=K_(T)+K_(R )=(1)/(2)MV^(2)+(1)/(2)Iomega^(2)` If .V. is the velocity when half the carpet has UNROLLED then as `V=(Romega)/(2)`, `M to (M)/(4)` and `I=(1)/(2)((M)/(4))((R )/(2))^(2)` `:.(7)/(8)MgR=(3)/(16)MV^(2)` on solving `V=((14gR)/(3))^((1)/(2))`