A cylinder of mass m rests in a carriage shown. The maximum...

A cylinder of mass $m$ rests in a carriage shown. The maximum acceleration of carriage so that the cylinder does not loose contact at $B$ is : (a) $3.66 \mathrm{~m} / \mathrm{s}^{2}$ (b) $10 \mathrm{~m} / \mathrm{s}^{2}$ (c) $45 \mathrm{~m} / \mathrm{s}^{2}$ (d) $8 \mathrm{~m} / \mathrm{s}^{2}$

A cylinder of mass $m$ rests in a carriage shown. The maximum acceleration of carriage so that the cylinder does not loose contact at $B$ is :
(a) $3.66 \mathrm{~m} / \mathrm{s}^{2}$
(b) $10 \mathrm{~m} / \mathrm{s}^{2}$
(c) $45 \mathrm{~m} / \mathrm{s}^{2}$
(d) $8 \mathrm{~m} / \mathrm{s}^{2}$

Key Concepts

Non-Inertial Reference Frames

This concept refers to analyzing motion from the perspective of an accelerating (non-inertial) frame, in which apparent forces—often called pseudo forces—must be introduced. In such frames, the usual equations of motion are modified to account for the effects of the acceleration of the reference frame, affecting how forces like gravity are perceived.

Effective Gravitational Acceleration

When an observer is in an accelerating system, the actual gravitational acceleration is combined with the pseudo acceleration due to the frame’s motion to create an effective gravitational field. This effective field governs the apparent weight and the net force acting on objects within that frame, altering the conditions for contact.

Normal Force and Contact Conditions

For an object to maintain contact with a surface, the normal force—the component of the contact force perpendicular to the surface—must remain greater than zero. At the point where the normal force is zero, the object loses contact. Analyzing when this condition occurs is essential to determine the maximum permissible acceleration in the system.

Curved Motion Dynamics

Dynamics on curved surfaces involves centripetal (or radial) acceleration that acts towards the center of the curve. At critical points, such as the top of a curved path, the combination of gravitational forces and the demands of centripetal acceleration may result in a situation where the normal force becomes zero, marking the threshold at which the object loses contact with the surface.

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