The period of roll Tr of a ship is determined by the formula:

$$T_{r}=\frac{2\pi K}{\sqrt{g\times GM}}$$

where,

• K is the radius of gyration (mass moment of inertia)
• g is the acceleration due to gravity, and
• GM is the metacentric height.

• The amplitude of roll, or the maximum angle of heel, does not affect the period of roll. While a larger amplitude implies a greater heeling moment and faster roll speed, the time taken to complete one full roll cycle remains constant for a given metacentric height and radius of gyration.

• The period of roll is directly proportional to the radius of gyration. A larger radius of gyration (indicating a greater distribution of mass further from the ship's centre of rotation) leads to a longer period of roll. Conversely, a smaller radius of gyration (mass concentrated closer to the centre) results in a shorter period. The distribution of cargo significantly impacts K; cargo concentrated centrally minimises K and the roll period, while dispersed cargo maximises K and the roll period.

• The period of roll is inversely proportional to the square root of the metacentric height (GM). A larger GM (a stiffer ship) leads to a shorter roll period, as the ship quickly returns to its upright position. A smaller GM (a tender ship) results in a longer roll period, with slower return to the upright.

The location of masses in the ship will effect the GM & K. So the period of roll will be affected.

• If masses are at bottom, G moves down, GM ↑, period of roll ↓.
• If masses are at top, G moves up, GM ↓, period of roll ↑.
• If masses are concentrated at centre, K ↓, period of roll ↓.
• If masses are away from centre, K ↑, period of roll ↑.