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 show

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 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

TestHub · Accepted Answer

In a continuous printing process suppose the total length of paper is $l$ .

If $r_{1}$ is the inner radius of roll then, $π r^{2} - π r_{1}^{2} = l \times b$              ...(i)
And at this moment, $v = r ω$ ...(ii)

Differentiating equation (i) w.r.t. time,
$\Rightarrow π \times 2 r \frac{d r}{d t} = \frac{d l}{d t} \cdot b$
$\Rightarrow \frac{d r}{d t} = \frac{v . b}{2 π r}$           ...(iii)

Differentiating equation (ii) w.r.t time
$\Rightarrow \frac{d v}{d t} = r (\frac{d ω}{d t}) + ω (\frac{d r}{d t})$
$⇒ 0 = r α + ω \frac{d r}{d t}$             $(∵ \frac{dr}{dt} is - ve)$
$⇒ \frac{d r}{d t} = \frac{r α}{ω}$
$⇒ α = \frac{ω}{r} (\frac{d r}{d t})$            ...(iv)

Using (iii) and (iv), $\frac{v b . ω}{2 π r^{2}} = \frac{v^{2} b}{2 π r^{3}} = α$

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