A train moving along a horizontal track. A pendulum...

A train moving along a horizontal track. A pendulum suspended from the roof makes an angle of $4^{\circ}$ with the vertical, The acceleration of the train is $\mathrm{ms}^{-2}\left(\mathrm{~g}=10 \mathrm{~ms}^{-2}\right)$
(A) $0.6$
(B) $0.7$
(C) $0.5$
(D) $0.2$

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Key Concepts

Non-Inertial Reference Frames

When analyzing motion from the perspective of an accelerating system, such as a train, a non-inertial reference frame is used. In these frames, fictitious forces must be introduced to account for the acceleration, affecting the apparent behavior of objects inside the frame, such as a pendulum.

Effective Gravity (Apparent Gravity)

In an accelerating frame, the combination of the actual gravitational force and the fictitious force due to acceleration results in an effective gravitational field, often referred to as apparent gravity. This effective gravity determines the equilibrium position of objects, leading to phenomena like the deflection of a pendulum’s string.

Force Balance and Equilibrium in Accelerating Systems

For an object at equilibrium in an accelerating frame, the net force acting on it must be zero. In the case of a pendulum, the horizontal fictitious force caused by acceleration and the vertical gravitational force combine such that the tangent of the deflection angle equals the ratio of the horizontal acceleration to gravitational acceleration. This principle allows one to calculate the acceleration based on the observed angle.

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