Question

The spool shown in the figure is placed on a rough horizontal surface and has inner radius r and outer radius R. The angle $\theta$ between the applied force and the horizontal can be varied. The critical angle $\left(\theta \right)$ for which the spool does not roll and remains stationary is given by :

• A. $\theta =co{s}^{-1}\left(\frac{r}{R}\right)$
• B. $\theta =co{s}^{-1}\left(\frac{2r}{R}\right)$
• C. $\theta =co{s}^{-1}\sqrt{\frac{r}{R}}$
• D. $\theta =si{n}^{-1}\left(\frac{r}{R}\right)$

Answer 1

The correct option is A $\theta =co{s}^{-1}\left(\frac{r}{R}\right)$

If the spool is not to translate,
$Fcos\theta =f$ ....(i)
If the spool is not to rotate,
$fR=Fr$ .....(ii)
or $\frac{fR}{r}cos\theta =f$
or $cos\theta =\frac{r}{R}\Rightarrow \theta =co{s}^{-1}\left(\frac{r}{R}\right)$
Also the line of action of F should pass through the bottom most point.