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 waters 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 consider the forces acting on the cone before and after it is fully submerged.
Before Submersion:
When the cone is above the water, there are two forces acting on it:
1. The force due to gravity pulling the cone downwards.
2. The buoyant force exerted by the water on the cone, which acts in the opposite direction of gravity.
Since the cone is initially at rest, the forces acting on it are balanced. Therefore, the magnitude of the buoyant force equals the magnitude of the downward gravitational force.
After Submersion:
When the cone is fully submerged, the buoyant force acting on it is equal to the weight of the displaced water, as per Archimedes' principle. The volume of the cone that is submerged is proportional to the ratio of its density to the density of the water.
We can set up the following equation to represent the forces acting on the cone after it is fully submerged:
Buoyant Force = Weight of the Displaced Water
ρ_water * V_submerged * g = ρ_cone * V_cone * g
Since the height of the cone is unknown, we can use similar triangles to find V_submerged/V_cone:
V_submerged/V_cone = h_submerged/h_cone
where h_submerged is the height of the submerged portion of the cone and h_cone is the total height of the cone.
Since the vertex of the cone is just touching the water's surface, h_submerged + h_cone = h_cone = 2r, where r is the radius of the base of the cone.
Now, let's calculate the ratio of the density of the cone to the density of the water:
ρ_cone/ρ_water = (V_submerged/V_cone)(ρ_water/ρ_water)
= (h_submerged/h_cone)(ρ_water/ρ_water)
= h_submerged/h_cone
The ratio of the density of the cone to the density of the water is equal to h_submerged/h_cone, where h_cone = 2r.
To find the value of h_submerged, we need to calculate it based on the given information that the cone has zero speed at the instant it becomes fully submerged. This implies that the work done by the buoyant force must be equal to the potential energy of the cone when it is fully submerged.
Potential Energy of the Cone = m * g * h_cone, where m is the mass of the cone.
Since the mass of the cone is proportional to its volume and density, we can rewrite the equation as:
Potential Energy of the Cone = ρ_cone * V_cone * g * h_cone
Since the potential energy is equal to the work done by the buoyant force, we have:
Work done by Buoyant Force = Potential Energy of the Cone
Buoyant Force * h_submerged = ρ_cone * V_cone * g * h_cone
Simplifying the equation:
ρ_water * V_submerged * g * h_submerged = ρ_cone * V_cone * g * h_cone
Since V_submerged/V_cone = h_submerged/h_cone, we substitute V_submerged = (h_submerged/h_cone) * V_cone:
ρ_water * (h_submerged/h_cone) * V_cone * g * h_submerged = ρ_cone * V_cone * g * h_cone
Canceling out g and V_cone:
ρ_water * h_submerged^2 = ρ_cone * h_cone^2
Substituting h_cone = 2r:
ρ_water * h_submerged^2 = ρ_cone * (2r)^2
Canceling out ρ_water:
h_submerged^2 = ρ_cone/ρ_water * 4r^2
Finally, taking the square root of both sides:
h_submerged = 2r * sqrt(ρ_cone/ρ_water)
Now that we have the value of h_submerged in terms of the ratio of the densities and the radius r, we can substitute it back into the ratio expression:
ρ_cone/ρ_water = h_submerged/h_cone
= 2r * sqrt(ρ_cone/ρ_water)/2r
= sqrt(ρ_cone/ρ_water)
To simplify this equation, let's square both sides:
(ρ_cone/ρ_water)^2 = ρ_cone/ρ_water
Now, cross-multiplying:
(ρ_cone/ρ_water)^2 - ρ_cone/ρ_water = 0
Factoring out (ρ_cone/ρ_water):
(ρ_cone/ρ_water)((ρ_cone/ρ_water) - 1) = 0
Since (ρ_cone/ρ_water) cannot be zero, we have:
(ρ_cone/ρ_water) - 1 = 0
ρ_cone/ρ_water = 1
Therefore, the ratio of the density of the cone to the density of the water is 1.