Questions: Principal Moments of Inertia and Principal Axes

Question 1

An engineer mounts a rotating shaft with an attached L-shaped bracket. Even though the shaft is balanced in the static sense (center of mass is on the axis), vibrations occur during rotation. What is the most likely cause?

• A. The shaft is rotating too slowly to achieve stable dynamics
• B. The rotation axis is not a principal axis, so angular momentum L is not parallel to ω, generating reaction torques
• C. Static balance guarantees dynamic balance, so the problem must be in the motor
• D. The bracket has too low a moment of inertia

Answer: (B)

Static balance (center of mass on the axis) does not guarantee dynamic balance. When the rotation axis is not a principal axis, the products of inertia are nonzero, meaning the inertia tensor is not diagonal in that frame. As a result, L = I·ω is not parallel to ω — the angular momentum vector is misaligned with the spin axis. This misalignment creates reaction torques that must be supplied by the bearings, producing vibration. Dynamic balancing requires aligning the rotation axis with a principal axis.

Question 2

A physicist tosses a rectangular book into the air trying to spin it about its three axes in turn. She finds it spins cleanly about the spine axis (smallest moment) and the cover-to-cover axis (largest moment), but tumbles chaotically when spun about the face-to-face axis (intermediate moment). What theorem explains this?

• A. The parallel-axis theorem — the intermediate axis has an incorrectly computed moment
• B. The intermediate axis theorem — rotation is dynamically unstable about the axis with the intermediate principal moment of inertia
• C. Conservation of angular momentum — angular momentum cannot be maintained about any axis without external torque
• D. Euler's equations — they only apply to axes with maximum or minimum moments

Answer: (B)

The intermediate axis theorem (also called the tennis racket theorem or Dzhanibekov effect) states that free rotation is dynamically stable only about the principal axes with maximum and minimum moments of inertia. A small perturbation away from the intermediate axis grows rather than decaying, leading to tumbling. This is a direct result of Euler's equations for torque-free rotation, and it has practical consequences for spacecraft attitude dynamics.

Question 3

For any rigid body, at least three mutually orthogonal principal axes always exist.

• T. True
• F. False

Answer: True

This follows from the spectral theorem for real symmetric matrices: the inertia tensor is a 3×3 real symmetric matrix and therefore always has three real eigenvalues (the principal moments) and three mutually orthogonal eigenvectors (the principal axes). This is a mathematical guarantee, regardless of the body's shape or mass distribution. For bodies with symmetry, principal axes may be apparent geometrically; for irregular bodies, they must be found by solving the eigenvalue problem.

Question 4

Rotation about a principal axis produces reaction torques in the bearings because the angular momentum is not aligned with the spin axis.

• T. True
• F. False

Answer: False

This is exactly backwards. Rotation about a principal axis is the special case where L and ω ARE parallel — no reaction torques are needed and no vibration is produced. Reaction torques arise when rotating about a non-principal axis, because the nonzero products of inertia cause L to be misaligned with ω. The principal axes are precisely the directions for which this problem disappears.

Question 5

Why is rotation about a principal axis dynamically 'clean,' and what happens physically when a body rotates about a non-principal axis?

Model answer: When rotating about a principal axis, the angular momentum vector L is parallel to the angular velocity ω. No reaction torques are needed to sustain the rotation, and the body spins without wobble. When rotating about a non-principal axis, the products of inertia are nonzero and L = I·ω points in a direction that differs from ω. This misalignment means the angular momentum vector continuously changes direction as the body rotates, requiring external torques (supplied by bearings) to sustain that change. These reaction forces manifest as vibration in machinery and as complex tumbling motion in free-spinning objects.

This is why engineering design cares about principal axes: rotating machinery should always spin about a principal axis to avoid dynamic loads on bearings. For spacecraft, attitude control systems must account for the intermediate axis instability. Finding principal axes reduces to an eigenvalue problem on the inertia tensor — a direct application of linear algebra to rigid body dynamics.