An object is kept on a smooth inclined plane of 1 in l. The...

An object is kept on a smooth inclined plane of 1 in $l$. The horizontal acceleration to be imparted to the inclined plane so that the object is stationary relative to the inclined is (a) $g \sqrt{l^{2}-1}$ (b) $g\left(l^{2}-1\right)$ (c) $\frac{g}{\sqrt{p^{2}}}$ (d) $\frac{g}{l^{2}-1}$

An object is kept on a smooth inclined plane of 1 in $l$. The horizontal acceleration to be imparted to the inclined plane so that the object is stationary relative to the inclined is
(a) $g \sqrt{l^{2}-1}$
(b) $g\left(l^{2}-1\right)$
(c) $\frac{g}{\sqrt{p^{2}}}$
(d) $\frac{g}{l^{2}-1}$

Key Concepts

Non-inertial Reference Frames

In dynamics, non-inertial frames are accelerating frames where objects appear to experience additional forces that do not arise from any physical interaction. In these frames, an object may seem to be at rest even if real forces are acting on it, provided the pseudo forces introduced by the frame's acceleration balance the other forces.

Pseudo Force

A pseudo force, also known as a fictitious force, arises when an observer is in a non-inertial frame of reference. It is introduced to account for the acceleration of the frame itself, allowing the observer to apply Newton’s laws as if in an inertial frame. Its magnitude is proportional to the mass of the object and the acceleration of the frame, but it acts in the opposite direction.

Force Resolution on an Inclined Plane

Analyzing forces on an inclined plane involves breaking down all acting forces into components parallel and perpendicular to the plane. This is particularly important when determining equilibrium conditions under the influence of gravitational forces and any additional accelerations, whether they arise from external inputs or the frame acceleration.

Equilibrium Analysis in Accelerated Systems

In problems where an object is required to remain stationary relative to a moving frame, equilibrium analysis is used by setting the net force to zero. This typically involves balancing gravitational components, any applied accelerations, and the corresponding pseudo forces, resulting in conditions that can be solved to find unknown accelerations or other quantities.

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