Q. The spool shown in figure is placed on rough horizontal surface and has the 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:-
A) `theta = cos^-1(r/R)`
The figure is below. The outer circle of radius `R` rolls on the rough surface while the force is applied to the inner circle making an angle `theta` with the horizontal.
The condition for static equilibrium is equality of moments and equality of forces on both horizontal and vertical axis.
The external applied force has the components
On the horizontal and vertical axis the equilibrium condition are
`F*cos(theta) =Ff` (1)
(The reaction `N` from the surface is not shown in the figure.)
The momentum equality is
`F*r = Ff*R`
which combined with (1) gives
`F*r = F*R*cos(theta)`
`cos(theta) = r/R`
Thus the angle for which the spool does not roll is
The correct answer is A) `theta =cos^(-1)(r/R)`