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A carpet of mass M made of inextensible material is r11 months agoShare 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 negligible small push is given to it. Calculate the horizontal velocity of the axis of the cylindrical part of the carpet when its radius reduces to R/2.11 months agoShare
11 months ago
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Dear Student,Please find below the solution to your problem.The rolling motion of the carpet is due to the lowering of the centre of mass of the carpet in unrolling.The potential energy thus released is converted into kinetic energy of the rolling part, the flat part being at rest.Initial potential energy: PE=MgRMass of the carpet of radius R/2 is proportional to M(0.5)^2 = M/4Hence, Potential energy at the point of interest is PEf =(M/4)g(R/2) = MgR/8Release of potential energy is Δ(PE) = mgR − MgR/8 = 7MgR/8The kinetic energy at this instant is given by KE = 0.5Mv2 + 0.5Iω2= 0.5*(M/4)*v^2 + 0.5∗(M/4)∗(R/2)^2/2*(v^2/(R/2)^2)= 3Mv^2/16∴3Mv^2/16 =7MgR/8⇒ v = √14gR/3Thanks and Regards
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