Animated Solution for Physics - Laws of Motion: A block is kept on a frictionless inclined surface with angle of inclination $\alpha$. The incline is given an acceleration $a$ to keep the block stationary. Then, $a$ is equal to

Keeping a Block Stationary on an Accelerating Incline

Imagine you are standing on a wedge that is suddenly accelerating to the right. If you place a block on it, your intuition tells you it should slide down the slope. But in this fascinating scenario, the wedge's acceleration is perfectly tuned to keep the block completely stationary relative to it. How does this happen? Let's dive into the physics and uncover the elegance of non-inertial frames.

The Non-Inertial Frame

To solve this problem elegantly, we need to change our perspective. Instead of watching the wedge accelerate from the ground, let's "jump" onto the wedge itself. By entering this accelerating frame of reference, we are now in a non-inertial frame.

Newton's laws of motion are strictly valid only in inertial (non-accelerating) frames. To make them work here, we must introduce a mathematical correction known as a pseudo force. Since the wedge accelerates to the right with an acceleration $a$, the pseudo force acts on the block in the exact opposite direction—to the left. The magnitude of this force is simply the mass of the block multiplied by the frame's acceleration:

Now that we are on the wedge, let's identify all the forces acting on the block: 1. Gravity ($mg$): Pulling straight down towards the center of the Earth. 2. Normal Force ($N$): Pushing perpendicularly outward from the surface of the incline. 3. Pseudo Force ($ma$): Pushing horizontally to the left.

Since the block is stationary on the incline, it is in perfect equilibrium in our non-inertial frame. The most strategic way to analyze this equilibrium is to resolve the forces parallel and perpendicular to the inclined surface.

Let's look at the forces acting parallel to the incline: - Gravity wants to pull the block down the slope. Its component along the incline is $mg\sin \alpha$. - The pseudo force, acting horizontally to the left, has a component that pushes the block up the slope. Using geometry, we find this component to be $ma\cos \alpha$.

For the block to remain perfectly still, the force trying to pull it down must be exactly balanced by the force pushing it up. This gives us our master equilibrium equation:

Notice something beautiful here? The mass $m$ appears on both sides of the equation. This means we can cancel it out entirely! The required acceleration does not depend on whether the block is a tiny pebble or a massive boulder.

Now, we simply isolate the acceleration $a$:

And there we have it! The exact acceleration required to keep the block from sliding is $g\tan \alpha$.

The Role of Friction

This problem assumes a perfectly frictionless surface, which means there is only one specific acceleration that works. But what if the surface was rough?

If friction were present, it would act to oppose any impending motion. If the wedge accelerated slightly slower than $g\tan \alpha$, friction would point up the incline to help the pseudo force. If the wedge accelerated slightly faster, friction would point down the incline to help gravity. This means that with friction, the block wouldn't just require one specific acceleration to stay put; instead, there would be a whole range of safe accelerations ($a_{\min }\le a\le a_{\max }$). Exploring that range is a fantastic exercise to deepen your understanding of mechanics!