Problem Statement
A solid wood cylinder is placed vertically in oil with the following properties:
- Diameter: 0.6m
- Height: 1.2m
- Specific gravity of wood: 0.6
- Specific gravity of oil: 0.85
Determine whether the cylinder is stable when floating vertically.
Solution
Calculate the Depth of Immersion (\(h\))
The weight of the cylinder is equal to the weight of the displaced oil: \[ \gamma_{\text{cylinder}} V_{\text{cylinder}} = \gamma_{\text{oil}} V_{\text{displaced oil}} \] \[ 0.6 \times 9810 \times \pi \times (0.3)^2 \times 1.2 = 0.85 \times 9810 \times \pi \times (0.3)^2 \times h \] Cancelling \( \pi \times (0.3)^2 \times 9810 \): \[ 0.6 \times 1.2 = 0.85 \times h \] \[ h = \frac{0.6 \times 1.2}{0.85} \] \[ h = 0.847 \text{ m} \]Calculate the Center of Buoyancy (\(OB\))
The center of buoyancy is at the centroid of the submerged volume: \[ OB = \frac{h}{2} = \frac{0.847}{2} \] \[ OB = 0.4235 \text{ m} \]Calculate the Center of Gravity (\(OG\))
Since the cylinder is uniform, its center of gravity is at the midpoint of its height: \[ OG = \frac{1.2}{2} \] \[ OG = 0.6 \text{ m} \] \[ BG = OG - OB = 0.6 - 0.4235 \] \[ BG = 0.1765 \text{ m} \]Calculate the Metacentric Height (\(GM\))
The metacentric radius (\(MB\)) is given by: \[ MB = \frac{I}{V} \] The moment of inertia about the vertical axis is: \[ I = \frac{1}{4} \pi r^4 \] \[ = \frac{1}{4} \pi (0.3)^4 \] \[ = \frac{1}{4} \pi \times 0.0081 = 0.00636 \text{ m}^4 \] The displaced volume: \[ V = \pi \times (0.3)^2 \times 0.847 \] \[ V = \pi \times 0.09 \times 0.847 \] \[ V = 0.2395 \text{ m}^3 \] \[ MB = \frac{0.00636}{0.2395} \] \[ MB = 0.0265 \text{ m} \] \[ GM = MB - BG \] \[ GM = 0.0265 - 0.1765 \] \[ GM = -0.15 \text{ m} \]Final Results:
- Depth of immersion: 0.847 m
- Metacentric height (\(GM\)): -0.15 m
- Since \(GM\) is negative, the cylinder is unstable.
Explanation
1. Stability Condition:
A floating body is stable if its metacentric height (\(GM\)) is positive. Here, \(GM\) is negative, meaning that the cylinder will tip over.
2. Why is the Cylinder Unstable?
- The center of gravity (\(OG\)) is higher than the center of buoyancy (\(OB\)), creating a high overturning moment.
- The metacentric height is negative, meaning the cylinder will not return to its vertical position if disturbed.
Physical Meaning
1. Ship and Boat Stability:
This principle is used to design ships and floating structures to ensure they do not capsize.