Consider a homogeneous right circular cylinder of length L, radius R, and specific gravity S, floating in water (S = 1) with its axis vertical. Show that the body is stable is R/L=[2S(1-S)]^(1/2).

Stability of a Floating Cylinder

Problem Statement

A homogeneous right circular cylinder of:

Length: \( L \)
Radius: \( R \)
Specific gravity: \( S \)

is floating in water with its axis vertical. Show that the body is stable if:

\[ \frac{R}{L} = \sqrt{2S(1 – S)} \]

Solution

1. Condition for Floating

The weight of the cylinder must equal the weight of the displaced water: \[ \gamma_{\text{cylinder}} V_{\text{cylinder}} = \gamma_{\text{water}} V_{\text{displaced water}} \] \[ S \times 9810 \times \pi R^2 L = 9810 \times \pi R^2 h \] Cancelling \( 9810 \pi R^2 \): \[ S L = h \]

2. Compute the Centers of Gravity and Buoyancy

The center of buoyancy (\( OB \)) is at the centroid of the submerged portion: \[ OB = \frac{h}{2} = \frac{S L}{2} \] The center of gravity (\( OG \)) for a uniform cylinder: \[ OG = \frac{L}{2} \] \[ BG = OG – OB = \frac{L}{2} – \frac{S L}{2} \] \[ BG = \frac{L}{2} (1 – S) \]

3. Compute the Metacentric Radius (\(BM\))

The metacentric radius (\( BM \)) is given by: \[ BM = \frac{I}{V} \] The moment of inertia about the vertical axis: \[ I = \frac{1}{4} \pi R^4 \] The displaced volume: \[ V = \pi R^2 h = \pi R^2 S L \] \[ BM = \frac{\frac{1}{4} \pi R^4}{\pi R^2 S L} \] \[ BM = \frac{R^2}{4 S L} \]

4. Compute the Metacentric Height (\(GM\))

\[ GM = BM – BG \] \[ GM = \frac{R^2}{4 S L} – \frac{L}{2} (1 – S) \] For stable equilibrium, \( GM \geq 0 \): \[ \frac{R^2}{4 S L} – \frac{L}{2} (1 – S) = 0 \] Solving for \( R/L \): \[ \frac{R^2}{4 S L} = \frac{L}{2} (1 – S) \] \[ \frac{R^2}{L^2} = 2 S (1 – S) \] \[ \frac{R}{L} = \sqrt{2 S (1 – S)} \]

Final Result:

For stable equilibrium, the ratio of \( R/L \) must satisfy:

\[ \frac{R}{L} = \sqrt{2 S (1 – S)} \]

Explanation

1. Stability Condition:
A floating body is stable if its metacentric height (\(GM\)) is positive. The metacentric height depends on the position of the center of buoyancy (\(B\)), the center of gravity (\(G\)), and the metacentric radius (\(BM\)).

2. Physical Interpretation:
– If \( R/L \) is too large, the cylinder may become unstable.
– The derived inequality ensures the cylinder remains stable and upright.
– This stability condition is important for designing floating cylindrical structures, such as offshore pipelines and buoys.

3. Effect of Relative Density \( S \):
– For small \( S \), the cylinder is very buoyant, requiring a smaller \( R/L \) for stability.
– For \( S \approx 1 \) (nearly as dense as water), the stability condition changes accordingly.

Physical Meaning

1. Engineering Applications:
– Used in designing floating cylindrical structures, such as buoys and offshore pipelines.
– Helps in stability analysis of floating tanks and pontoon bridges.

2. Industrial and Real-World Uses:
– Used for designing floating storage tanks.
– Ensures stability in cylindrical underwater research modules.

Consider a homogeneous right circular cylinder of length L, radius R, and specific gravity S, floating in water (S = 1) with its axis vertical. Show that the body is stable is .