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Question

In a continuous printing process, paper roll is wrapped on a cylindrical core which is free to rotate about a fixed horizontal axis. The paper is drawn into the process at a constant speed v, as shown in the figure. If r is the radius of the paper on the roll at any given time and b is the thickness of the paper, then the angular acceleration of the roll at this instant is


A
v2b2π2r3
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B
Zero because v is constant
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C
v2b2πr3
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D
Cannot be calculated from the given information
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Solution

The correct option is C v2b2πr3
In a continuous printing process, suppose the total length of paper on the roll is l. If r1 is the inner radius of the roll, then
πr2πr21=l×b ...(1)
And at this moment, v=rω ...(2)
[where v is the speed at which paper is drawn into the process]


Differentiating equation (1) w.r.t time,
π×2rdrdt0=b×dldt
or drdt=vb2πr ....(3)
[dldt=v i.e length of the paper on the roll is decreasing]
Differentiating equation (2) w.r.t time
dvdt=r(dωdt)+ω(drdt)
0=rα+ωdrdt
[v=constant]
drdt=rαω
α=ωr(drdt) ....(4)

Using (3) and (4)
α=ωr×(vb2πr)
=v2b2πr3 [v=wr]

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