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Chapter 9: Problem 31

(a) What is the buoyant force on \(0.90 \mathrm{kg}\) of ice floating freely in liquid water? (b) What is the buoyant force on \(0.90 \mathrm{kg}\) of ice held completely submerged under water?

Short Answer

Expert verified
Q: Find the buoyant force acting on 0.90 kg of ice floating in liquid water and when the same ice is submerged completely underwater. A: The buoyant force acting on 0.90 kg of ice floating freely in liquid water and when the same ice is submerged completely underwater is the same and equal to the weight of the displaced water, which can be calculated using the steps provided in the solution.

Step by step solution

01

Calculate the volume of the ice

To calculate the volume of the ice, we can use its mass and density following the formula: Volume = Mass / Density The mass of the ice is \(0.90 \mathrm{kg}\), and the density of ice is \(920 \mathrm{kg/m^3}\). Therefore, the volume of the ice is: \(V_{ice} = \frac{0.90}{920} \mathrm{m^3}\)
02

Calculate the weight of the displaced water

Since the ice displaces an equal amount of water, we can now calculate the weight of the water displaced when submerged. To do this, we need to find the mass of the displaced water and then multiply by the acceleration due to gravity (\(g = 9.81 \mathrm{m/s^2}\)). We first find the mass of the displaced water by multiplying the volume of the ice by the density of the water: Mass of displaced water = Volume of ice × Density of water \(M_{water} = V_{ice} × 1000 \mathrm{kg/m^3}\) Now, we can find the weight of the displaced water by multiplying the mass with the acceleration due to gravity: \(W_{water} = M_{water} × 9.81 \mathrm{N}\)
03

Find the buoyant force

According to Archimedes' principle, the buoyant force is equal to the weight of the displaced water. Thus: \(F_{buoyant} = W_{water}\) Now, we can calculate the buoyant force for both cases (a) and (b). Since the ice displaces an equal amount of water in both cases, the buoyant force will have the same value. Calculating: \(a)\) \(F_{buoyant\,a} = W_{water}\) \(b)\) \(F_{buoyant\,b} = W_{water}\) So the buoyant force acting on \(0.90 \mathrm{kg}\) of ice floating freely in liquid water and when the same ice is submerged completely underwater is the same and equal to the weight of the displaced water, which can be calculated using the steps provided.

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