You need to feed your pet crab who lives under water. To do you, you will attach the food (which we assume has 0 mass) to the bottom of a cylindrical buoy dropped (at rest) from a height of 0.5 meters. The buoy must be made from a material which has 1/3rd the density of water. What should the proportions of the buoy be (height and radius) in order for the oscillation of the buoy to just barely make it down 0.4 meters below the water to reach your crab? Assume g=10

Question

You need to feed your pet crab who lives under water. To do you, you will attach the food (which we assume has 0 mass) to the bottom of a cylindrical buoy dropped (at rest) from a height of 0.5 meters. The buoy must be made from a material which has 1/3rd the density of water. What should the proportions of the buoy be (height and radius) in order for the oscillation of the buoy to just barely make it down 0.4 meters below the water to reach your crab? Assume g=10

Answer 1

To calculate the proportions of the buoy, we need to consider the buoyancy force and the weight of the buoy.

First, let's calculate the buoyancy force acting on the buoy. The buoyancy force is equal to the weight of the water displaced by the buoy, which can be calculated using Archimedes' principle. The weight of the water displaced is the volume of the water displaced multiplied by the density of water and the acceleration due to gravity (g).

Since the buoy must sink to a depth of 0.4 meters below the water surface, the volume of water displaced can be calculated as the area of the base of the buoy times the depth submerged:

Volume of water displaced = base area × depth submerged

We can find the area of the base using the formula for the area of a circle: πr^2, where r is the radius of the buoy's base.

Next, let's calculate the weight of the buoy. The weight is equal to the mass of the buoy multiplied by the acceleration due to gravity:

Weight of the buoy = mass of the buoy × g

The mass of the buoy can be calculated using the formula:

Mass of the buoy = density of the material × volume of the buoy

Given that the material has 1/3rd the density of water, the density of the material is 1/3 × density of water.

To create an equilibrium situation where the buoy just barely reaches the crab, the buoyancy force must counterbalance the weight of the buoy. Mathematically, we can express this equilibrium condition as:

Buoyancy force = Weight of the buoy

Using the calculations above, we can set up an equation to find the proportions of the buoy:

(1/3 × density of water) × (πr^2) × 0.4 × g = (density of the material) × ((1/3 × density of water) × (πr^2) × h) × g

Simplifying the equation, we find:

0.4 = h

This tells us that the height of the buoy should be equal to the depth submerged, which is 0.4 meters. The radius of the buoy, however, does not affect the equilibrium condition since it cancels out on both sides of the equation. Hence, we can choose any value for the radius as long as the height is 0.4 meters.

In conclusion, to just barely reach the crab that is situated 0.4 meters below the water, the proportions of the buoy can vary greatly in terms of radius, but the height of the buoy should be equal to 0.4 meters.