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

`Fh =F*cos(theta)`

`F_v =F*sin(theta)`

On the horizontal and vertical axis the equilibrium condition are

`F*cos(theta) =Ff`   (1)

`F*sin(theta)=N`

(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)`

or equivalent

`cos(theta) = r/R`

Thus the angle for which the spool does not roll is

`theta =arccos(r/R)`

The correct answer is A) `theta =cos^(-1)(r/R)`

Images:

Image (1 of 1)