A solid right circular cone of uniform mass density is initially at rest above a body of water, so that its vertex is just touching the water's surface with its axis of symmetry along the vertical.
Now, the cone falls into the water, and has zero speed at the instant it becomes fully submerged. What is the ratio of the density of the cone to the density of the water? Submit your answer to 2 decimal places.
Details and Assumptions:
There is an ambient downward gravitational field.
Assume that the buoyant force is the only force exerted by the water on the cone.
To find the ratio of the density of the cone to the density of the water, we need to use Archimedes' principle, which states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
Let's follow these steps to find the solution:
Step 1: Define the variables.
Let's assume:
- Density of the cone: ρ(cone)
- Density of the water: ρ(water)
- Volume of the cone: V(cone)
- Mass of the cone: m(cone)
- Volume of water displaced: V(displaced)
- Mass of water displaced: m(displaced)
- Acceleration due to gravity: g
Step 2: Calculate the volume and mass of the cone.
The volume of a right circular cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. Since the vertex is just touching the water's surface, the height of the cone is equal to the depth of the water it displaces.
Step 3: Calculate the volume and mass of the water displaced.
Since the cone is fully submerged and is at rest, the volume of water displaced is equal to the volume of the cone, V(displaced) = V(cone). Using the density of water, we can then calculate the mass of water displaced, m(displaced) = V(displaced) * ρ(water).
Step 4: Calculate the net force acting on the cone.
The net force acting on the cone is given by the difference between the weight of the cone and the buoyant force acting on it. The weight of the cone is equal to mass times gravity, F(weight) = m(cone) * g. The buoyant force is equal to the weight of the water displaced, F(buoyant) = m(displaced) * g.
Step 5: Set up and solve the equation for the net force.
Using the principle of equilibrium, the net force acting on the cone is zero. Therefore, F(net) = F(weight) - F(buoyant) = 0.
Step 6: Rearrange and solve for the ratio of densities.
Rearrange the equation F(net) = 0 to find the ratio of densities:
m(cone) * g - m(displaced) * g = 0
m(cone) = m(displaced)
ρ(cone) * V(cone) = ρ(water) * V(displaced)
Divide both sides by V(cone) * ρ(water) to obtain the ratio of densities:
ρ(cone) / ρ(water) = V(displaced) / V(cone)
Step 7: Substitute the known values and calculate the ratio of densities.
Plug in the values and calculate the ratio using the formula from Step 6.
Remember to round the final answer to 2 decimal places, as requested.
Note: Depending on the specific values provided in the question, you may need additional information, such as the height or dimensions of the cone, to calculate the ratio of densities accurately.