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),Mto(M)/(4)ANDI=(1)/(2)((M)/(4))((R)/(2))^(2)` `:.(7)/(8)MgR=(3)/(16)MV^(2)` on SOLVING `V=((14gR)/(3))^(1/2)`