The spool shown in the figure is placed on a rough horizontal surface and has inner radius `r` and outer radius `R`. The angle `theta` between the applied force and the horizontal can be varied. The critical angle `theta` for which the spool does not roll and remains stationary is given by

Options

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

Answer

A

Explanation

If the spool is not to translate
`Fcostheta=f`…………i
If the spool is not to rotate
`Fr=Ffr`….ii
or `(fR)/rcostheta=f`
or `costheta=r/Rimplies theta=cos^(-1)(r/R)`
Also the line of action of `F` should pass through the bottom most point.