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

Answer 1

The correct option is D ${\cos }^{-1}\left(\frac{r}{R}\right)$


Let $F=$ applied force and $f=$ force of friction (left wards)
As spool is in equilibrium,
$\therefore F\cos \theta =f.....\left(1\right)$
Net torque about centre $=0$
$F.r=fR$
$\Rightarrow \frac{F}{R}=\frac{f}{r}\Rightarrow \frac{F}{f}=\frac{R}{r}$
$\Rightarrow \frac{f}{F}=\frac{r}{R}......\left(2\right)$
From equation (1) and (2)
$\cos \theta =\frac{r}{R}$
$\Rightarrow \theta ={\cos }^{-1}\left(\frac{r}{R}\right)$