Λrectangular container of base dimensions 2 m ×1 m and height 0.5 m is filled with water upto

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Λrectangular container of base dimensions 2 m ×1 m and...

$\Lambda$ rectangular container of base dimensions $2 \mathrm{~m} \times 1 \mathrm{~m}$ and height $0.5 \mathrm{~m}$ is filled with water upto height of $0.2 \mathrm{~m}$. 'The mass of the empty container is $1 \mathrm{~kg}$. The coefficient of friction between conLainer and plane is $0.5$. The container is placed on the pane inclined at $45^{\circ}$ to the horizontal. The angle of the water surface to the horizontal is (a) $\tan ^{\prime}\left(\frac{1}{2}\right)$ (b) $\tan ^{1} 2$ (c) $\tan ^{\prime}\left(\frac{1}{3}\right)$ (d) $\tan ^{1} 3$

$\Lambda$ rectangular container of base dimensions $2 \mathrm{~m} \times 1 \mathrm{~m}$ and height $0.5 \mathrm{~m}$ is filled with water upto height of $0.2 \mathrm{~m}$. 'The mass of the empty container is $1 \mathrm{~kg}$. The coefficient of friction between conLainer and plane is $0.5$. The container is placed on the pane inclined at $45^{\circ}$ to the horizontal. The angle of the water surface to the horizontal is
(a) $\tan ^{\prime}\left(\frac{1}{2}\right)$
(b) $\tan ^{1} 2$
(c) $\tan ^{\prime}\left(\frac{1}{3}\right)$
(d) $\tan ^{1} 3$

Key Concepts

Vector Addition of Accelerations

Determining the effective acceleration in a system where multiple accelerations are present – such as gravitational acceleration and the acceleration due to motion – requires vector addition. The net (or effective) acceleration, which acts on the fluid in the container’s non?inertial frame, is found by combining these individual accelerations. The direction of this net acceleration dictates the orientation of the free surface, as the surface must be perpendicular to it.

Free Surface Equilibrium in Accelerated Fluids

In any fluid at equilibrium, the free surface adjusts itself so that it is always perpendicular to the net acceleration acting on the fluid. In a non?inertial (accelerating) frame, this net acceleration is the vector sum of the gravitational acceleration and any additional inertial (pseudo) acceleration due to the motion of the container. This concept explains why, when a container is accelerating, the liquid’s free surface appears tilted relative to the container even though it remains horizontal in an inertial frame.

Non?Inertial Reference Frames and Pseudo Forces

When analyzing motion from the accelerating frame of a container, it is necessary to include pseudo forces that effectively ‘act’ in the opposite direction of the acceleration. These forces allow us to treat the container as if it were at rest and to determine the effective gravitational field acting on the liquid. The use of pseudo forces simplifies the analysis by providing a way to ‘correct’ for the acceleration of the container.

Friction and Motion on an Inclined Plane

A body on an inclined plane experiences a component of gravitational force along the plane that tends to cause it to slide. The frictional force, which depends on the coefficient of friction and the normal force, opposes this motion. In scenarios where friction is insufficient to balance the downhill component of gravity, the body accelerates, and the magnitude of this acceleration is determined by the difference between the gravitational component along the slope and the maximum frictional force.

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