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 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

Step-by-step solution

In a continuous printing process suppose the total length of paper is l . If r 1 is the inner radius of roll then, π r 2 - π r 1 2 = l × b ...(i) And at this moment, v = r ω ...(ii) Differentiating equation (i) w.r.t. time, ⇒ π × 2 r d r d t = d l d t · b ⇒ d r d t = v . b 2 π r ...(iii) Differentiating equation (ii) w.r.t time ⇒ d v d t = r d ω d t + ω d r d t ⇒ 0 = r α + ω d r d t ∵ dr dt is - ve ⇒ d r d t = r α ω ⇒ α = ω r d r d t ...(iv) Using (iii) and (iv), v b . ω 2 π r 2 = v 2 b 2 π r 3 = α