A massless spool of inner radius `r` outer radius `R` is placed against a vertical wall and a titled split floor as shown. A light inextensible thread is tightly wound around the spool through which a mass `m` is hainging. There exists no friction at point `A`, while the coefficient of friction between the spool and point `B` is `mu`. The angle between the two surface is `theta`

Options

A. the magnitude of force on the spool at B in order to maintain equilibrium is mg`sqrt((r/R)^(2)+(1-r/R)^(2)1/(tan^(2)thetha)`
B. the magnitude of force on the spool at B in order to maintain equilibrium is mg`(1-r/R)1/(tan thetha)`
C. the minimum value of mu for the system to remain in equilibrium is`(cot theta)/((R//r)-1)`
D. the minimum value of mu for the system to remain in equilibrium is `(tan thetha)/((R//r)-1`

Answer

A, D

Explanation

`mgr=fR.....(1)`
`N_(1)sintheta+f=mg......(2)`
`N_(1)costheta=N_(2)......(3)`
`f lt muN_(2)....(4)`