A disc is free ot 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 of the ribboned disc be `R` at any time and thickness of the ribbon ve `d(ltltR)`. Then the force `(F)` required to pull the ribbon as a funcation of radius `R` is.

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))`