If a solid conical buoy of height H and relative density S floats in water with axis vertical and apex upwards, show that the height above the water surface of the conical buoy is equal to H(1-S)^(1/3).

Problem Statement

A solid conical buoy of height \( H \) and relative density \( S \) floats in water with its axis vertical and apex upwards.

Show that the height above the water surface of the conical buoy is given by:

\[ y = H (1 – S)^{\frac{1}{3}} \]

Solution

1. Define Variables

\( R \) = Radius of the cone at its base
\( r \) = Radius of the cone at the water surface
\( y \) = Height of the cone above the water
\( H \) = Total height of the cone
\( S \) = Relative density of the cone

2. Apply Buoyancy Principle

The weight of the cone must be equal to the weight of the water displaced:

\[ \gamma_{\text{cone}} V_{\text{cone}} = \gamma_{\text{water}} V_{\text{displaced}} \] \[ S \gamma_{\text{water}} \left(\frac{1}{3} \pi R^2 H\right) = \gamma_{\text{water}} \left(\frac{1}{3} \pi R^2 H – \frac{1}{3} \pi r^2 y\right) \] \[ R^2 S H = R^2 H – r^2 y \] \[ y = \frac{R^2}{r^2} H (1 – S) \quad \text{(Equation 1)} \]

3. Use Similar Triangles

From the geometry of the cone, using similar triangles:

\[ \frac{R}{r} = \frac{H}{y} \] \[ R = \frac{r H}{y} \]

4. Solve for \( y \)

Substituting into Equation (1):

\[ y = \frac{H^2}{y^2} H (1 – S) \] \[ y^3 = H^3 (1 – S) \] \[ y = H (1 – S)^{\frac{1}{3}} \]

Final Result:

The height of the buoy above the water surface is:

\[ y = H (1 – S)^{\frac{1}{3}} \]

Explanation

1. Buoyancy Principle:
A floating object displaces its own weight in water. This means that the weight of the conical buoy must equal the weight of the water it displaces.

2. Volume Considerations:
The total volume of the cone is given by the formula for the volume of a cone, and the submerged portion of the volume is determined by removing the part above water.

3. Geometry and Proportions:
By applying the concept of similar triangles, we establish a relationship between the radii and heights, allowing us to express everything in terms of the height of the buoy above water.

4. Final Derivation:
Solving for \( y \) gives the final result, showing that the height of the buoy above water depends on the relative density of the buoy material.

Physical Meaning

1. Effect of Density:
The higher the relative density \( S \), the more of the buoy is submerged. If \( S = 1 \) (same as water), the buoy is fully submerged and does not float.

2. Buoy Shape Influence:
Since the buoy is conical, its submerged volume and corresponding buoyant force change with depth, affecting how much of it remains above water.

3. Engineering Applications:
This principle is used in the design of floating devices, including buoys, boats, and offshore structures, ensuring stability and proper flotation.

If a solid conical buoy of height H and relative density S floats in water with axis vertical and apex upwards, show that the height above the water surface of the conical buoy is equal to H(1-S)^(1/3).