In a continuous printing process paper is drawn into the press at a constant speed v. Denoting by 'r' the radius of the paper roll at any given time and by 'b' the thickness of the paper. What is the angular acceleraion of the paper roll?

Options

A. `(bv^(2))/(2pir^(3))`
B. `(2bv^(2))/(pir^(3))`
C. `(bv^(2))/(4pir^(3))`
D. `(4bv^(2))/(pir^(3))`

Answer

A

Explanation

Let one layer of paper be unrolled
`vDeltat =2pir and Deltar=-b`
`(Deltar)/(Deltat)=(-bv)/(2pir)=(dr)/(dt)`
`alpha=(domega)/(dt)=(d)/(dt)((v)/(r ))`
`=(1)/(r)(dv)/(dt)+v(d)/(dt)((1)/(r))`
`= 0 -(v)/(r^(2))(dr)/(dt)`
`=(-(v)/(r^(2)))(-(bv)/(2pir))`
`(bv^(2))/(2pir^(3))`
`alpha=(bv^(2))/(2pir^(3)`