A solid cone of specific gravity 0.7 floats in water with its apex downwards. Determine the least apex angle of the cone for equilibrium.

Problem Statement

A solid cone with the following properties is floating in water with its apex downwards:

Specific gravity: 0.7

Determine the least apex angle (\(2\theta\)) required for the cone to remain in stable equilibrium.

Solution

1. Calculate the Depth of Immersion (\(h\))

The weight of the cone must equal the weight of the displaced water: \[ \gamma_{\text{cone}} V_{\text{cone}} = \gamma_{\text{water}} V_{\text{displaced water}} \] \[ 0.7 \times 9810 \times \frac{1}{3} \pi R^2 H = 9810 \times \frac{1}{3} \pi r^2 h \] \[ h = \frac{0.7 R^2 H}{r^2} \] Substituting \( R = H \tan\theta \) and \( r = h \tan\theta \): \[ h = \frac{0.7 (H \tan\theta)^2 H}{(h \tan\theta)^2} \] \[ h = 0.8879H \]

2. Calculate the Centers of Gravity and Buoyancy

The center of gravity of the cone: \[ OG = \frac{3}{4} H = 0.75H \] The center of buoyancy is at the centroid of the submerged volume: \[ OB = \frac{3}{4} h = 0.75 \times 0.8879H = 0.6659H \] \[ BG = OG – OB = 0.75H – 0.6659H = 0.0841H \]

3. Calculate the Metacentric Height (\(GM\))

The metacentric radius (\(MB\)) is given by: \[ MB = \frac{I}{V} \] The moment of inertia about the vertical axis: \[ I = \frac{1}{4} \pi r^4 \] The displaced volume: \[ V = \frac{1}{3} \pi r^2 h \] \[ MB = \frac{\frac{1}{4} \pi r^4}{\frac{1}{3} \pi r^2 h} \] \[ MB = \frac{0.75 r^2}{h} \] Substituting \( r = h \tan\theta \): \[ MB = 0.75 h \tan^2\theta \] \[ MB = 0.75 \times 0.8879H \tan^2\theta = 0.6659H \tan^2\theta \] \[ GM = MB – BG \] \[ GM = 0.6659H \tan^2\theta – 0.0841H \] For stability, \( GM > 0 \): \[ 0.6659H \tan^2\theta – 0.0841H > 0 \] \[ \tan^2\theta > 0.1263 \] \[ \tan\theta > 0.3553 \] \[ \theta > 19.56^\circ \] \[ 2\theta = 2 \times 19.56 = 39.12^\circ \]

Final Result:

Least apex angle (\(2\theta\)): 39.12°

Explanation

1. Stability Condition:
A floating cone is stable if its metacentric height (\(GM\)) is positive. This requires that the restoring moment is greater than the overturning moment.

2. Role of the Apex Angle:
– The apex angle determines how much of the cone is submerged.
– A larger apex angle increases the waterplane area, improving stability.

3. Importance of Stability Analysis:
– Ensures floating structures do not tip over.
– Used in shipbuilding and offshore platform design.

Physical Meaning

1. Engineering Applications:
The stability of conical floating objects is used in designing buoys, submarines, and underwater research platforms.

2. Industrial and Real-World Uses:
– Helps in designing floating cone-shaped tanks.
– Used in buoyancy-driven underwater vehicles.

A solid cone of specific gravity 0.7 floats in water with its apex downwards. Determine the least apex angle of the cone for equilibrium.