A blumb bob is hung from the ceiling of a train compartment....

A blumb bob is hung from the ceiling of a train compartment. The train moves on an inclined track of inclination $30^{\circ}$ with horizontal. Acceleration of train up the plane is $a=9 / 2$. The angle which the string supporting the bob makes with normal to the ceiling in equilibrium is (a) $30^{\circ}$ (b) $\tan ^{-1}\left(\frac{2}{\sqrt{3}}\right)$ (c) $\tan ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ (d) $\tan ^{-1}(2)$

A blumb bob is hung from the ceiling of a train compartment. The train moves on an inclined track of inclination $30^{\circ}$ with horizontal. Acceleration of train up the plane is $a=9 / 2$. The angle which the string supporting the bob makes with normal to the ceiling in equilibrium is
(a) $30^{\circ}$
(b) $\tan ^{-1}\left(\frac{2}{\sqrt{3}}\right)$ (c) $\tan ^{-1}\left(\frac{\sqrt{3}}{2}\right)$
(d) $\tan ^{-1}(2)$

Key Concepts

Equilibrium in Accelerating Systems

When an object, such as a pendulum, is in an accelerating system, it attains equilibrium when the forces (including both real and pseudo forces) balance. The equilibrium configuration is determined by the vector sum of gravitational forces and acceleration-induced pseudo forces, which together dictate the direction of the effective gravitational pull.

Vector Decomposition of Forces

Vector decomposition involves breaking a force down into its component parts along chosen axes. This technique is crucial in problems involving inclined planes or accelerating frames since it allows the analysis of forces in directions parallel and perpendicular to a surface or the acceleration, aiding in the determination of equilibrium conditions.

Non-Inertial Reference Frames

Non-inertial reference frames are accelerating frames of reference in which observers experience additional forces (known as pseudo or fictitious forces). These forces must be introduced to account for the observed deviations from Newton’s laws, allowing for proper analysis of motion and equilibrium in a non-inertial context.

Effective Gravity

Effective gravity is the net acceleration acting on a body, resulting from the combination of true gravitational acceleration and the acceleration of the reference frame. In scenarios where the frame itself accelerates, effective gravity redefines the apparent direction and magnitude of the gravitational force perceived by an object.

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