An open vessel containing water is given a constant...

An open vessel containing water is given a constant acceleration $a$ in the horizontal direction. Then the free surface of water gets sloped with the horizontal at an angle $\theta$ given by (A) $\theta=\tan ^{-1}\left(\frac{a}{g}\right)$ (B) $\theta=\tan ^{-1}\left(\frac{g}{a}\right)$ (C) $\theta=\sin ^{-1}\left(\frac{a}{g}\right)$ (D) $\theta=\cos ^{-1}\left(\frac{g}{a}\right)$

An open vessel containing water is given a constant acceleration $a$ in the horizontal direction. Then the free surface of water gets sloped with the horizontal at an angle $\theta$ given by
(A) $\theta=\tan ^{-1}\left(\frac{a}{g}\right)$
(B) $\theta=\tan ^{-1}\left(\frac{g}{a}\right)$
(C) $\theta=\sin ^{-1}\left(\frac{a}{g}\right)$
(D) $\theta=\cos ^{-1}\left(\frac{g}{a}\right)$

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Key Concepts

Non-Inertial Reference Frames

This concept involves analyzing physical phenomena from frames that are accelerating relative to an inertial frame. Within such frames, fictitious or inertial forces must be introduced to account for the acceleration when applying Newton's laws. These pseudo-forces directly affect the observed behavior of objects and fluids, making them essential in understanding dynamics in accelerating systems.

Effective Gravitational Field

When a system is subjected to acceleration, the net or effective gravitational field is the vector sum of the actual gravitational field and the acceleration-induced field. This combined field determines the new direction in which objects and fluids will 'feel' pulled, influencing the orientation of surfaces in equilibrium, such as the free surface of a fluid.

Hydrostatic Equilibrium

In fluids at rest, or in a state of equilibrium, the free surface aligns perpendicular to the net force acting on the fluid. This means that in an accelerated frame, the free surface of a liquid will adjust its orientation so that it remains perpendicular to the effective gravitational field, ensuring that the pressure is constant along the surface.

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