The spool shown in the figure is placed on a | vedclass

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(A) For the spool to remain stationary,it must be in both translational and rotational equilibrium.$1$. Translational Equilibrium: The net force on th

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$	heta = \cos^{-1} \left( \frac{r}{R} ight)$

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(A) For the spool to remain stationary,it must be in both translational and rotational equilibrium.$1$. Translational Equilibrium: The net force on the spool must be zero. Considering the horizontal forces,the applied force component $F \cos 	heta$ must be balanced by the static frictional force $f_s$ acting at the point of contact with the ground.$f_s = F \cos 	heta$$2$. Rotational Equilibrium: The net torque about the point of contact with the ground must be zero. The force $F$ is applied at a distance $r$ from the center. The torque due to force $F$ about the point of contact is $F 	imes r$. For the spool to not rotate,this must be balanced by the torque of the force components. However,a simpler way is to take the torque about the center of the spool. The torque due to friction $f_s$ at the contact point (distance $R$ from the center) must balance the torque due to the force $F$ (distance $r$ from the center).$f_s 	imes R = F 	imes r$Substituting $f_s = F \cos 	heta$ into the torque equation:$(F \cos 	heta) 	imes R = F 	imes r$Dividing both sides by $F 	imes R$:$\cos 	heta = \frac{r}{R}$Therefore,the critical angle is:$	heta = \cos^{-1} \left( \frac{r}{R} ight)$

The spool shown in the figure is placed on a rough horizontal surface and has an inner radius $r$ and an 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:

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