An air-conditioning system requires a 20-m-long section of 15-cm diameter duct work to be laid u

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An air-conditioning system requires a 20-m-long section of...

An air-conditioning system requires a 20-m-long section of $15-\mathrm{cm}$ diameter duct work to be laid underwater. Determine the upward force the water will exert on the duct. Take the densities of air and water to be $1.3 \mathrm{~kg} / \mathrm{m}^3$ and 1000 $\mathrm{kg} / \mathrm{m}^3$, respectively.

An air-conditioning system requires a 20-m-long section of $15-\mathrm{cm}$ diameter duct work to be laid underwater.
Determine the upward force the water will exert on the duct. Take the densities of air and water to be $1.3 \mathrm{~kg} / \mathrm{m}^3$ and 1000 $\mathrm{kg} / \mathrm{m}^3$, respectively.

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Key Concepts

Volume Calculation for Cylindrical Objects

Calculating the volume of a cylinder, which is given by the formula V = ?r²h (where r is the radius and h is the height or length), is essential when determining the amount of fluid displaced by a cylindrical object. This method is widely used in engineering and physics applications involving cylindrical shapes.

Fluid Density

Fluid density, defined as the mass per unit volume of a fluid, plays a crucial role in determining the weight of the displaced fluid and, consequently, the magnitude of the buoyant force. High-density fluids like water result in a larger buoyant force compared to lower-density fluids such as air.

Archimedes' Principle

Archimedes' Principle states that an object submerged in a fluid is buoyed upward by a force equal to the weight of the fluid that the object displaces. This principle is fundamental in understanding buoyancy and explains why objects in fluid appear lighter than in air.

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an immersed object. It can be calculated using the product of the fluid's density, the gravitational acceleration, and the volume of the displaced fluid. This concept is key in problems involving flotation and submersion.

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