Ship rolling: a floating ship has radius of gyration about the waterline k = 4 m and metacentric height GM = 72.5 cm. Its small-angle rolling period is closest to which value?

Introduction / Context:For small-amplitude rolling of a floating body, the restoring moment is proportional to the metacentric height GM, yielding a simple harmonic oscillator. The period depends on the radius of gyration about the waterline and the hydrostatic stiffness m * g * GM.

Given Data / Assumptions:

• Radius of gyration k = 4 m.
• Metacentric height GM = 72.5 cm = 0.725 m.
• Small-angle (linear) oscillations; g ≈ 9.81 m/s^2.
• Ship mass cancels from the period expression.

Concept / Approach:

The roll period T is T = 2 * π * sqrt( I / (m * g * GM) ), where I = m * k^2. Cancelling m gives T = 2 * π * sqrt( k^2 / (g * GM) ).

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Order-of-magnitude check: larger GM shortens period; here GM ~ 0.7 m reasonable for a mid-sized vessel, giving a few π seconds—consistent with 3π.

Why Other Options Are Wrong:

π and 2π are too short (would require much larger GM); 4π is too long. π/2 is unphysical for a ship of this size.

Common Pitfalls (misconceptions, mistakes):

Forgetting to convert GM to metres; using diameter instead of radius of gyration; omitting the 2π factor typical of SHM periods.

Final Answer:

3π