A small block of mass m lying at rest at point P of a wedge...

A small block of mass $m$ lying at rest at point $P$ of a wedge having a smooth semi-circular track of radius
$R$. What should be the minimum value of horizontal acceleration $a_{0}$ of wedge so that mass can just reach the point $Q$ ?
(A) $g / 2$
(B) $\sqrt{g}$
(C) $g$
(D) Not possible

Key Concepts

Non?Inertial Reference Frames

When analyzing systems in accelerating frames, such as a wedge with a horizontal acceleration, it is important to account for the fact that the frame is non?inertial. In these frames, fictitious or pseudo forces appear due to the acceleration of the frame itself, which must be included in the analysis to correctly predict the behavior of objects within the frame.

Pseudo Forces

In a non-inertial frame, the block experiences a pseudo force opposite the direction of the wedge's acceleration. This concept is crucial because the pseudo force modifies the effective gravitational field experienced by the block, and thereby influences its motion along the curved path of the wedge.

Energy Conservation in Conservative Systems

Even in non-inertial frames, when the forces involved are conservative (including the effective forces created by the wedge's acceleration), energy conservation can be applied. This means that the sum of the kinetic, gravitational, and pseudo potential energies remains constant, allowing one to determine the minimal conditions required for the block to reach a certain position.

Dynamics on Curved Surfaces

The block moves along a smooth, curved path provided by the semi-circular track, where the component of the effective force tangential to the track determines its motion. Understanding the dynamics on a curved surface involves resolving forces along the path and applying energy methods or Newton’s laws to assess whether the block has enough energy to overcome changes in the effective potential energy.

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