Routh’s rule for a symmetric body: A solid body is symmetrical about two perpendicular semi-axes. Using Routh’s rule, the moment of inertia about an axis through the centre of gravity is given by m*(a^2 + b^2)/n. What is the value of n?

Introduction / Context:
Routh’s rule (a compact statement of standard results) provides formulas for the second moment of mass (moment of inertia) of ellipsoids and other symmetric bodies. It generalizes familiar results such as I = (2/5) m r^2 for a solid sphere and I = (1/5) m (b^2 + c^2) for a solid ellipsoid about a principal axis.

Given Data / Assumptions:

• Homogeneous, rigid body.
• Symmetry about two perpendicular semi-axes of lengths a and b.
• Axis passes through the centre of gravity and is a principal axis.

Concept / Approach:
For a solid ellipsoid with semi-axes a, b, c, the principal moments are: I_x = (1/5) m (b^2 + c^2), I_y = (1/5) m (a^2 + c^2), I_z = (1/5) m (a^2 + b^2). Routh’s compact form expresses each as m*(sum of squares of the two semi-axes perpendicular to the moment axis) divided by 5.

Step-by-Step Solution:

Verification / Alternative check:
For a sphere (a = b = c = r), I = (2/5) m r^2, which corresponds to m*(r^2 + r^2)/5 = (2/5) m r^2.

Why Other Options Are Wrong:

• 2, 3, 4, 6: None match the established principal moment formulas for ellipsoids; only 5 reproduces the canonical results.

Common Pitfalls:
Confusing diameters with semi-axes; using thin-shell formulas instead of solid-body formulas; forgetting the factor 1/5 for homogeneous solids.

Final Answer:
5