Question

An air-conditioning system requires a 35 -m-long section of 15 -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.

Short answer

Answer: The upward force exerted on the duct by the water is approximately 6054.375 N.

Step-by-step solution

Convert the given dimensions to standard units

Firstly, convert the diameter of the duct to meters: Diameter = 15 cm = 0.15 m Length = 35 m Now, we have the dimensions in standard units.

Calculate the Volume of the duct

The duct is in cylindrical shape, so we will use the formula for the volume of a cylinder: \(V = \pi r^2 h\) Where: \(V\) = Volume of the duct \(r\) = radius of the duct (half of the diameter) \(h\) = length of the duct We will now plug in the values to calculate the volume: \(V = \pi (0.075)^2 (35)\) \(V \approx 0.6175 \;\text{m}^3\)

Calculate the weight of displaced water

Next, we will use the volume of the duct to determine the mass of the water displaced by the duct. The formula for this is: \(W_\text{water} = \rho_\text{water} \cdot V\) Where: \(W_\text{water}\) = weight of displaced water \(\rho_\text{water}\) = density of water (given as 1000 kg/m³) Now, plug in the values to calculate the weight of the displaced water: \(W_\text{water} \approx 1000 \cdot 0.6175\) \(W_\text{water} \approx 617.5 \;\text{kg}\)

Calculate the buoyancy force (Upward force)

Using Archimedes' principle, the buoyancy force (upward force) on the duct can be calculated by multiplying the weight of the displaced water (found in step 3) by the acceleration due to gravity (g ≈ 9.81 m/s²): \(F_\text{upward} = W_\text{water} \cdot g\) \(F_\text{upward} \approx 617.5 \cdot 9.81\) \(F_\text{upward} \approx 6054.375 \;\text{N}\) Now, we have found the upward force exerted on the duct by the water to be approximately 6054.375 N.

Key concepts

Archimedes' Principle

When it comes to understanding why objects float or sink in a fluid, Archimedes' principle is fundamental. According to this principle, the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

Let's put it in a real-world context: imagine you're in a pool and you push a ball under the water. The ball pops back up because the water is pushing it up with a force equal to the weight of the water the ball pushed aside. This is Archimedes' principle in action.

So, when our duct is submerged, we need to figure out how much water it's displacing to find the upward force, also known as the buoyant force. Keep this in mind whenever you're dealing with problems related to objects in a fluid, because Archimedes' principle doesn't just tell us something will float or sink, but also quantifies the buoyant force involved.

Density and Buoyancy

The concepts of density and buoyancy are intertwined. Density is mass per unit volume and it's a property that indicates how much stuff is packed into a certain space. It's usually represented with the Greek letter rho \(\rho\).

When we talk about buoyancy, we are referring to an object's ability to float in a fluid. This ability is directly connected to how the object's density compares to the fluid's density. If the object is less dense than the fluid, it will float. If it's more dense, it will sink.

In our case, the displaced water's weight is directly related to the water's density \(\rho_{water}\) and the volume of water displaced by the duct. Since the duct's contents are less dense than the water (air, in this case), it experiences an upward buoyancy force. This explains why objects like ships, made of steel which is denser than water, can float: they are not solid steel but contain a lot of air-filled space, making the overall density less than that of water.

Cylindrical Volume Calculation

Calculating the volume of a cylinder involves using a specific formula \(V = \pi r^2 h\), where \(V\) represents the volume, \(r\) is the radius of the circular base, and \(h\) is the height, or length, of the cylinder.

For the duct in our exercise, we first convert its diameter to meters to get the radius, as the radius is half the diameter. With the radius and the length of the duct, applying the cylindrical volume formula gives us the volume of space the duct occupies.

This volume calculation is crucial because it tells us how much water is being displaced when the duct is fully submerged. And according to Archimedes' principle, this is what we need to find the buoyant force. By keeping the formula and its components in mind, solving related problems becomes more manageable, whether you're estimating the capacity of a pipe or determining the displacement of a ship.