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 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 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)I OMEGA^(2)` If .V. is the velocity when half the carpet has unrolled then as `V=(R omega)/(2), "" M to (M)/(4)` and `I=(1)/(2)((M)/(4))((R )/(2))^(2)` `therefore (7)/(8)MgR = (3)/(16) MV^(2)` on solving `V=((14gR)/(3))^((1)/(2))`