1. Set up the problem in the accelerating frame

A block is on a frictionless incline of angle $\alpha$. The incline is accelerated horizontally with acceleration $a$ such that the block remains stationary relative to the incline.

To analyze this, it is easiest to work in the non-inertial frame of the incline.

In this frame, the block is at rest, so we include a pseudo force of magnitude $ma$ on the block opposite to the acceleration of the incline.

2. Forces acting on the block

The forces on the block are:

• Weight: $mg$ vertically downward
• Normal reaction: $N$ perpendicular to the incline
• Pseudo force: $ma$ horizontally opposite to the acceleration of the incline

Since the surface is frictionless, there is no friction.

3. Condition for the block to remain stationary on the incline

For the block to remain at rest relative to the incline, the net force along the plane must be zero.

Resolve forces along the incline:

• Component of gravity down the incline: $mg\sin\alpha$
• Component of pseudo force up the incline: $ma\cos\alpha$

For equilibrium along the plane, $ma\cos\alpha = mg\sin\alpha$

Cancel $m$: $a\cos\alpha = g\sin\alpha$

Hence, $a = g\tan\alpha$

4. Match with options

$a = g\tan\alpha$

So the correct option is:

C: $g\tan\alpha$

5. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So they agree.