I Became Fan Of Calculus After I Solved This Olympiad Level Physics Problem | Kalda 15

In this Physics video in Hindi for the chapter : "Properties of Fluids" of Class 11, we discussed problem no. 15 from the handout 'Problems on Mechanics' by Jaan Kalda. This problem presents a fascinating situation involving a hemispherical container placed upside down on a perfectly smooth horizontal surface. Water is poured into the container through a small hole situated at what becomes the topmost point once the hemisphere is inverted. The key moment arrives when the container becomes completely full, and at that precise instant, water begins to leak from the narrow gap between the container’s rim and the table. The question asks us to determine the mass of the container, given the density of water and the radius of the hemispherical shape. This elegantly structured problem showcases the typical style of Jaan Kalda, where simple physical setups lead to deep insights rooted in the chapter "Properties of Fluids" and demand conceptual reasoning at the level expected in IIT-JEE Advanced, INPhO and IPhO preparation.

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Kalda's Problem 15 : Solved With Only 1 Simple Concept
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To solve this question, we analyze how water inside the hemisphere exerts pressure on every part of its curved inner surface. As more water is poured in, the resultant upward forces caused by fluid pressure start acting on the hemispherical walls. Because the container is resting on a smooth surface, it relies solely on its weight and the pressure distribution to maintain contact with the table. The moment water begins leaking identifies the instant at which the normal reaction at the edge becomes zero. By studying this condition of equilibrium, one can deduce the exact mass the container must have to counteract the upward force exerted by the water. This method combines principles of fluid statics and equilibrium, reflecting the intellectual flavor found across Jaan Kalda’s problems in the handout "Problems on Mechanics."

Definition 1 : Fluid Pressure.
Fluid pressure is the perpendicular force per unit area exerted by a fluid on any surface with which it is in contact. In this problem, fluid pressure plays a central role because the distribution of pressure over the curved surface of the hemispherical container generates upward forces that influence the container’s equilibrium.

Definition 2 : Buoyant Effect Without Immersion.
This definition refers to the upward resultant force generated inside a cavity due to fluid pressure acting on inner surfaces, even when the object itself is not submerged in the fluid. Here, the hemispherical container experiences an upward force arising from the water pushing against its interior surface.

Theorem 1 : Condition for Vertical Equilibrium.
This theorem states that for a body to remain in vertical equilibrium, the total upward forces acting on it must balance the total downward forces. This principle allows us to determine the exact moment at which the normal reaction at the rim becomes zero.

Theorem 2 : Hydrostatic Force Distribution Principle.
This principle states that the force exerted by a fluid at rest acts perpendicular to the containing surface and increases with depth. This concept helps us understand how the upward force resulting from water pressure varies across the curved surface of the hemisphere.

Throughout this discussion, focus remains firmly on the conceptual foundations of the chapter "Properties of Fluids," enabling students to appreciate how fluid pressure, reaction forces, and equilibrium conditions combine in complex yet beautifully structured problems. This problem by Jaan Kalda highlights the depth of reasoning that IIT-JEE Advanced aspirants and physics olympiad students must master. As we explore the logic step-by-step, the video reinforces essential principles linked to fluid mechanics while strengthening problem-solving intuition. This makes the content exceptionally valuable for learners targeting IIT-JEE Advanced and those using Jaan Kalda’s materials for higher-level physics training.
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