A solid cylinder, \(1 \mathrm{~m}\) diameter and \(800 \mathrm{~mm}\) high, is of uniform relative density \(0.85\) and floats with its axis vertical in still water. Calculate the periodic time of small angular oscillations about a horizontal axis.

When an object is floating, it experiences an upward force called buoyancy. This force is equal to the weight of the fluid the object displaces.
In this exercise, since the cylinder floats in water, the buoyant force is equal to the weight of the water displaced by the cylinder.

Buoyancy can be represented mathematically as:
\[F_b = \rho_{fluid} \times V_{displaced} \times g \]
Where:
- \( F_b \) is the buoyant force
- \( \rho_{fluid} \) is the density of the fluid
- \( V_{displaced} \) is the volume of the fluid displaced
- \( g \) is the acceleration due to gravity

For our cylinder, we calculate the volume of the water displaced using the cylinder's dimensions and relative density.

Metacentric height (GM) is a measure of the stability of a floating object. It is the distance between the center of gravity (G) of the object and the metacenter (M).
To calculate the metacentric height, we use the formula:
\[ GM = \frac{I}{V} - BG \]
Where:
- \( I \) is the moment of inertia of the water plane area about the axis of rotation.
- \( V \) is the volume of water displaced
- \( BG \) is the distance between the center of buoyancy (B) and the center of gravity (G)
The center of buoyancy is located at the center of the volume of displaced water. By computing these values, we can determine the metacentric height, which in turn indicates how stable the floating cylinder is.

Angular oscillations refer to the movement of the cylinder around a horizontal axis. Imagine the cylinder tilting back and forth in the water; these are small angular oscillations.

The frequency and stability of these oscillations are influenced by the metacentric height. A higher metacentric height means the cylinder will return to a stable position more quickly after tilting.
Angular oscillations are small deviations and typically studied under the assumption of small angles, enabling linear approximations.

The periodic time, denoted as T, is the time it takes for the cylinder to complete one full cycle of oscillation.
To find this, we use the formula:
\[ T = 2\bigpi \frac {K}{\text{GM} \times g} \]
Where:
- \( K \) is a constant, typically considered as unity for small oscillations
- \( GM \) is the metacentric height
- \( g \) is the acceleration due to gravity
Plugging in the calculated values of GM and the known gravitational acceleration, we can determine the periodic time for the small angular oscillations of the cylinder, revealing how quickly it oscillates back and forth in water.