A disc is free to rotate about an axis passing through its centre and perpendicular to its plane. The moment of inertia of the disc about its rotation axis is I. A light ribbon is tightly wrapped over it in multiple layers. The end of the ribbon is pulled out at a constant speed of u. Let the radius of the ribboned disc be R at any time and thickness of the ribbon be `d (lt lt R)`. Find the force (F) required to pull the ribbon as a function of radius R.

Options

A. Angular acceleration of the disc at any insant is proportional to `1/(R^(3))`, where `R` is radius of the ribboned disc
B. Angular acceleration o the disc at any instant is proportional to `1/(R^(2))` where `R` is the radius of the ribboned disc.
C. Force required to pull the ribbon at constant speed `u` is `F=(Iu^(2)d)/(2piR^(4))`
D. Force required to pull the ribbon at constant speed `u` is `F=(Iu^(2)d)/(piR^(4))`

Answer

A, C

Explanation

`omegaR=u=`constant
`omega(dR)/(dt)+R(domega)/(dt)=0`
`:.(domega)/(dt)=-(omega)/R(dR)/(dt)`
In time `dt` radius decreases by `dR`
`-2pi RdR= uddt`
`:.(dR)/(dt)=-(ud)/(2piR)`
`(domega)/(dt)=alpha=(omega)/R (ud)/(2piR)`
`alpha=(u^(2)d)/(2piR^(3))`
`tau=(Iu^(2)d)/(2piR^(3))`
`:.F=(Iu^(2)d)/(2piR^(4))`