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A carpet of mass M made of inextensible ...

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 ar rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is given to it. Calculate the horizontal velocity of the axis of the cylindrical part of the carpet when its rdius reduces to `R//2`.

Text Solution

Verified by Experts

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)`
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