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

• A. ${60}^{\circ }$
• B. ${45}^{\circ }$
• C. ${20}^{\circ }$
• D. ${30}^{\circ }$

Answer 1

The correct option is B ${45}^{\circ }$

Let $F$ is the applied force and $f$ is the induced frictional force.

Since, the spool is at rest even after applying force $F$.

So, we can balance the component of applied force with the induced frictional force $f$ in the horizontal direction.

So, we can write the force balance equation along horizontal direction as,

$F\cos \theta =f$

$\therefore \cos \theta =\frac{f}{F}.....\left(1\right)$

Since the spool is at rest and also not rotating, so the total torque about the centre is zero,
$\overrightarrow{\tau }=0........\left(2\right)$

$[\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}=rF\sin \delta \hat{\eta }]$
Since, here $\delta$ is ${90}^{\circ }$.

So, from $\left(2\right)$, we can write,
$\Rightarrow 1\times F-\sqrt{2}\times f=0$

$\Rightarrow \frac{f}{F}=\frac{1}{\sqrt{2}}....\left(3\right)$

From $\left(1\right)$ and $\left(3\right)$,

$\cos \theta =\frac{1}{\sqrt{2}}$

$\therefore \theta ={45}^{\circ }$

Hence, option $\left(b\right)$ is correct answer.