To what depth will a 2.4m diameter log 5m long and sp gr 0.4 sink in fresh water?

Depth of Floatation of a Log in Water

Problem Statement

A cylindrical log has the following properties:

Diameter: 2.4m
Length: 5m
Specific gravity: 0.4

Determine the depth to which the log will sink in fresh water.

Solution

1. Define Parameters

\[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{2.4}{2} = 1.2 \text{ m} \] Depth of floatation (\( DC \)) = ?

2. Apply the Buoyancy Principle

The weight of the log must be equal to the weight of the displaced water: \[ \gamma_{\text{log}} V_{\text{log}} = \gamma_{\text{water}} V_{\text{displaced water}} \] \[ 0.4 \times 9810 \times \pi \times (1.2)^2 \times 5 = 9810 [V_{\text{sector OABC}} – V_{\text{2 triangles}}] \]

3. Substitute Values

\[ 9810 \times 9.04 = 9810 \left[ \frac{2\theta}{360} \times \pi \times (1.2)^2 \times 5 – 2 \times 0.5 \times 1.2 \cos\theta \times 1.2 \sin\theta \right] \] \[ 9.04 = 0.1256\theta – 0.72 \sin 2\theta \]

4. Solve for \( \theta \)

Using trial and error: \[ \theta = 80^\circ, \quad \text{Right side} = 9.8 \] \[ \theta = 75^\circ, \quad \text{Right side} = 9.06 \] \[ \theta = 74.9^\circ, \quad \text{Right side} = 9.04 \] \[ \Rightarrow \theta = 74.9^\circ \]

5. Calculate Depth of Floatation (\( DC \))

\[ DC = OC – OD \] \[ = 1.2 – 1.2 \sin 74.9 \] \[ = 0.041 \text{ m} \]

Final Result:

Depth of floatation: 0.041 m

Explanation

1. Floating Equilibrium:
The log floats because its density is less than that of water. It sinks until the weight of the displaced water equals its own weight.

2. Circular Segment Calculation:
Since the log is partially submerged, we use a sector of the circle minus the two small right-angled triangles to determine the submerged volume.

3. Trial and Error for \( \theta \):
Since the equation involves trigonometric terms, we solve it iteratively by testing different values of \( \theta \) until we get the correct balance.

4. Depth Calculation:
Once \( \theta \) is known, we determine the vertical distance from the center of the log to the waterline, giving the depth of floatation.

Physical Meaning

1. Practical Application in Logging:
Logs are transported by floating them on rivers. Knowing how much of a log remains submerged helps in estimating buoyancy and stability.

2. Importance in Marine Engineering:
Similar calculations are used in designing floating platforms, boats, and ships to ensure they remain stable while partially submerged.

3. Stability Considerations:
A log that is too deeply submerged may roll over or sink if additional weight is applied. Engineers use these calculations to design floating structures that remain stable in water.