The spool shown in figure is placed on rough horizontal surface a… - Vidyadip

MCQ

The spool shown in figure is placed on 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 $(\theta )$ for which the spool does not roll and remains stationary is given by

✓
$\theta = cos^{-1} \left( {\frac{r}{R}} \right)$
B
$\theta = cos^{-1}$$\left( {\frac{{2r}}{R}} \right)$
C
$\theta = cos^{-1}$$\sqrt {\frac{r}{R}} $
D
$\theta = sin^{-1}$$\left( {\frac{r}{R}} \right)$

✓

Answer

Correct option: A.

$\theta = cos^{-1} \left( {\frac{r}{R}} \right)$

a
If the spool does not have translational motion, net force on it is zero.

For the horizontal direction,

$F \cos \theta=F_{f r}$

If the spool does not rotate, net torque is zero.

$F_{f r} R=F r$

$\Rightarrow F \cos \theta R=F r$

$\therefore \cos \theta=\frac{r}{R}$

$\Rightarrow \theta=\cos ^{-1} \frac{r}{R}$

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