Question

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

• A. ${\cos }^{-1}\left(\frac{R}{r}\right)$
• B. ${\sin }^{-1}\left(\sqrt{1-\frac{r^2}{R^2}}\right)$
• C. $\pi -{\cos }^{-1}\left(\frac{r}{R}\right)$
• D. ${\sin }^{-1}\left(\frac{r}{R}\right)$

Answer 1

Answer: (B)

${\sin }^{-1}\left(\sqrt{1-\frac{r^2}{R^2}}\right)$

Let $F=$ the applied force and $f=$ force of friction (leftwards)
Net horizontal force $=0$ $\therefore F\cos \theta =f.....\left(i\right)$
Net torque about centre $=0$
$F\cdot r=fR$ Substituting $\frac{f}{R}=\frac{r}{R}$ in Eq.(I) we have,
$\cos \theta =\frac{r}{R}or\sin \theta =\sqrt{1-\frac{r^2}{R^2}}$
$\therefore \theta ={\sin }^{-1}\left(\sqrt{1-\frac{r^2}{R^2}}\right)$