A spool is pulled at an angle `theta` with the horizontal on a rough horizontal surface as shown in the figure. If the spool remains at rest, the angle `theta` is equal to

Options

A. `cos^(-1)((R)/(r))`
B. `sin^(-1)(sqrt(1-(r^(2))/(R^(2))))`
C. `pi-cos^(-1)((r)/(R))`
D. `sin^(-1)((r)/(R))`

Answer

B

Explanation

Let `F=` the applied force and `f=` force of friction (leftwards)
Net horizontal force`=0`
`thereforeFcostheta=f`
Net torque about centre `=0`
`therforeF.r=fR`
Substituting `(f)/(R)=(r)/(R)` in Eq. (i) we have
`costheta=(r)/(R)`
or `sintheta=sqrt(1-(r^(2))/(R^(2)))`
`thereforetheta=sin^(-1)(sqrt(1-(r^(2))/(R^(2))))`