Surprizing One Line Solution For This Complex "Water Leakage" Problem from Jaan Kalda - 15

Описание: In this Physics video in Hindi for the chapters : "Laws of Motion" and "Properties of Fluids" of Class 11, we discussed problem no. 15 from the handout 'Problems on Mechanics' by Jaan Kalda. In this discussion, we explore a fascinating situation involving a hemispherical container and poured water.

The question states :-
A hemispherical container is placed upside down on a smooth horizontal surface. Through a small hole at the bottom of the container, water is then poured in. Exactly when the container gets full, water starts leaking from between the table and the edge of the container. Find the mass of the container if water has density ρ and radius of the hemisphere is R.


This question is typical of Jaan Kalda’s elegant mechanics style, where simple physical setups lead to deeply insightful applications of the principles taught in the chapter "Laws of Motion" and "Properties of Fluids". The step-by-step reasoning in the solution helps students preparing for IIT-JEE Advanced and physics olympiads understand the interplay between forces, pressure, and reaction at constraints.

To solve this question, we analyse how water filled inside the hemisphere exerts pressure on all parts of the container including the circular rim resting on the table. Since the table is smooth, the only way the container remains in equilibrium is through the balance of vertical forces transmitted through the water and the weight of the container. The moment water begins to flow out from under the rim tells us that the normal reaction at that point just becomes zero. This condition allows us to find the mass of the container by equating the downward forces with the upward fluid forces. The idea behind the solution showcases how pressure distribution can create lifting tendencies and how equilibrium conditions help determine unknown physical quantities. This problem helps reinforce the foundational understanding required for IIT-JEE Advanced level problem solving.

Definition 1 : Pressure.
Pressure is a physical quantity that represents the force exerted perpendicularly on a surface by a fluid in contact with it. It helps describe how fluids transmit forces uniformly in all directions and is essential in understanding how water inside the hemispherical container pushes on its curved surface.

Definition 2 : Normal Reaction.
Normal reaction is the perpendicular contact force exerted by a surface to support an object placed on it. In this question, the normal reaction from different parts of the circular rim adjusts itself depending on the pressure distribution created by the water inside the container. When leakage begins, the normal reaction at a particular point becomes zero, signifying a critical force-balance condition.

Theorem 1 : Condition of Vertical Force Equilibrium.
This theorem states that for any system in translational equilibrium in the vertical direction, the sum of all upward forces must be equal to the sum of all downward forces. This principle is crucial in determining the exact point where the container is just about to lift at its rim.

Theorem 2 : Hydrostatic Force Distribution Principle.
This principle states that a fluid at rest exerts force perpendicular to the surface in contact, and the magnitude of this force depends on the depth of the fluid. This principle is used to calculate the upward forces acting on the hemispherical wall due to the water inside it, which ultimately affects the equilibrium of the container.

Through this problem by Jaan Kalda, students gain a deeper appreciation of how "Laws of Motion" and "Properties of Fluids" combines with fluid pressure concepts to solve challenging mechanics questions. The reasoning involved reflects the style of physics required for IIT-JEE Advanced and olympiad-level problem solving. The chapter "Laws of Motion" and "Properties of Fluids" is repeatedly highlighted here because mastering these ideas is essential for developing strong conceptual foundations. This video is especially useful for learners aiming for IIT-JEE Advanced and olympiads like INPhO, and the logical structure presented in this Jaan Kalda problem will enhance your analytical skills.
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