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

Let F= the applied force and f= force of friction (leftwards) 

Net horizontal force =0

$\therefore \mathrm{Fcos\theta }=\mathrm{f}$

Net torque about centre =0

$\therefore \mathrm{F}·\mathrm{r}=\mathrm{fR}$

Substituting $\frac{\mathrm{f}}{\mathrm{R}}=\frac{\mathrm{r}}{\mathrm{R}}$ in Eq. (i) we have,

$$\begin{gathered} \mathrm{cos\theta }=\frac{\mathrm{r}}{\mathrm{R}} \\ \mathrm{sin\theta }=\sqrt{1-\frac{{\mathrm{r}}^2}{{\mathrm{R}}^2}} \\ \mathrm{\theta }={\sin }^{-1}\left(\sqrt{1-\frac{{\mathrm{r}}^2}{{\mathrm{R}}^2}}\right) \end{gathered}$$