Quiz:Sliding motion along an inclined plane

See sliding motion along an inclined plane for background information.

Force diagram and acceleration analysis

1 Consider a situation where ${\mu }_s=m{u}_k=\mu$ is the coefficient of static as well as kinetic friction between the block and the incline. What can we say about the normal force between the block (of mass $m$) and the incline as a function of the angle of incline $\theta$ (this is the angle that the incline makes with the horizontal)?

	The magnitude of normal force equals $mg$ and is independent of $\theta$
	The magnitude of normal force decreases from $mg$ to 0 as $\theta$ increases from 0 to $\pi /2$
	The magnitude of normal force increases from 0 to $mg$ as $\theta$ increases from 0 to $\pi /2$
	The magnitude of normal force increases for $\theta$ increasing from 0 to ${\tan }^{-1}(\mu )$ and decreases for $\theta$ increasing from ${\tan }^{-1}(\mu )$ to $\pi /2$
	The magnitude of normal force decreases for $\theta$ increasing from 0 to ${\tan }^{-1}(\mu )$ and increases for $\theta$ increasing from ${\tan }^{-1}(\mu )$ to $\pi /2$

2 Consider a situation where ${\mu }_s={\mu }_k=\mu$ is the coefficient of static as well as kinetic friction between the block and the incline. Once the angle of the incline (denoted $\theta$) exceeds ${\tan }^{-1}({\mu }_s)$, the magnitude of downward acceleration $a$ is an increasing function of $\theta$. What is its derivative with respect to $\theta$?

	$g(\sin \theta +\mu \cos \theta )$
	$g(\sin \theta -\mu \cos \theta )$
	$g(\cos \theta -\mu \sin \theta )$
	$g(\cos \theta +\mu \sin \theta )$