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Grade: 11

                        

A carpet of mass M made of inextensible material is r 11 months ago Share 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 ago Share

4 years ago

Answers : (2)

Ashutosh Mohan Sharma
askIITians Faculty
180 Points
							509-1092_Screen Shot 2016-11-18 at 12.00.38 PM.png
						
4 years ago
Rishi Sharma
askIITians Faculty
614 Points
							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=MgR
Mass of the carpet of radius R/2​ is proportional to M(0.5​)^2 = M/4​
Hence, Potential energy at the point of interest is PEf ​=(M/4​)g(R/2​) = MgR/8​
Release of potential energy is Δ(PE) = mgR − MgR/8​ = 7​MgR/8
The kinetic energy at this instant is given by KE = 0.5​Mv2 + 0.5​Iω2
= 0.5*​(M/4)*​v^2 + 0.5​∗(M/4)​∗(R/2​)^2/2*​(v^2/(R/2​)^2)​
= 3​Mv^2/16
∴3​Mv^2/16 =7​MgR/8
⇒ v = √14​gR/3​

Thanks and Regards
4 months ago
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