Routh’s Rule for Bodies Symmetrical About a Perpendicular Axis For a spherical (or axisymmetric) body of mass M, let S be the sum of squares of the two semi-axes perpendicular to the axis considered. According to Routh’s rule, what is the mass moment of inertia about the centroidal axis?

Introduction / Context:
Routh’s rule provides compact expressions for the mass moment of inertia of ellipsoids or bodies symmetrical about a perpendicular axis. For such a body, the inertia about one principal axis depends on the squares of the other two semi-axes.

Given Data / Assumptions:

• Mass M, centroidal principal axes.
• Axisymmetry about a perpendicular axis; S is the sum of squares of the two semi-axes perpendicular to the axis considered.
• Uniform mass distribution.

Concept / Approach:
For an ellipsoid with semi-axes a, b, c, the mass moments are I_x = (M/5)(b^2 + c^2), I_y = (M/5)(c^2 + a^2), I_z = (M/5)(a^2 + b^2). Writing S as the sum of the squares of the two semi-axes perpendicular to the chosen axis leads directly to I = (M/5) S.

Step-by-Step Solution:

Verification / Alternative check:
For a solid sphere (a = b = c = r), S = r^2 + r^2 = 2 r^2 ⇒ I = (M/5) * 2 r^2 = (2/5) M r^2, the well-known result for a solid sphere about any diameter.

Why Other Options Are Wrong:
M S / 3 and M S / 4 would overpredict inertia and do not match standard ellipsoid formulas; “None of these” is incorrect because M S / 5 is the exact expression.

Common Pitfalls:
Confusing area moments with mass moments; forgetting that S excludes the semi-axis along the chosen axis.

Final Answer:
M S / 5.