Relative Equilibrium of Liquids - Examples & Applications

Solved Problems

Example

An open rectangular tank

3 \text{ m}

long,

2 \text{ m}

wide, and

2 \text{ m}

deep is filled with water to a depth of

1.5 \text{ m}

. If the tank is accelerated horizontally along its length at

2.5 \text{ m/s}^2

, determine the maximum pressure at the bottom of the tank and whether any water spills.

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Example

An open cylindrical tank

1.2 \text{ m}

in diameter and

1.8 \text{ m}

deep is filled with water to a depth of

1.2 \text{ m}

. If the tank is rotated about its vertical axis at

60 \text{ rpm}

, determine the depth of water at the center and at the wall.

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Example

Problem 3: Liquid Spillage from an Accelerating Tank An open rectangular tank

3 \text{ m}

long and

2 \text{ m}

high is filled with water to a depth of

1.5 \text{ m}

. If the tank is accelerated horizontally along its length at

2.5 \text{ m/s}^2

, how much water is spilled?

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Example

Problem 4: Rotating Cylinder Forced Vortex An open cylindrical tank

1 \text{ m}

in diameter and

2 \text{ m}

high is filled with water to a depth of

1.2 \text{ m}

. It is rotated about its vertical axis at

45 \text{ rpm}

. Determine the height of the paraboloid formed.

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Case Study 1: Centrifugal Casting

Context: Centrifugal casting is a manufacturing process that uses the principles of a forced vortex.

Application: Molten metal is poured into a rapidly spinning cylindrical mold. The centrifugal forces push the denser liquid metal against the mold walls, forming a hollow cylinder as it solidifies. The forced vortex ensures impurities and lighter slag are forced toward the inner surface, which can later be machined away, leaving a dense, high-quality cast part like a pipe or cylinder liner.

Case Study 2: Fuel Tank Design in Aircraft

Context: Aircraft experience varying accelerations during flight.

Application: During takeoff, landing, and maneuvers, aircraft fuel tanks undergo linear acceleration, causing the fuel surface to tilt. Designers must place fuel pickup points (sumps) strategically to ensure a continuous supply to the engines, even when the fuel level is low and the surface is steeply inclined. Baffles are often installed inside tanks to minimize the sloshing effect and maintain relative equilibrium more effectively.

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