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title: "A student calculates the rotational inertia \\(I_A\\) of a rigid body of mass \\(M\\) about an axis \\(A\\) located a distance \\(d_A\\) from the center of mass. To determine the rotational inertia \\(I_B\\) about a parallel axis \\(B\\) located a distance \\(d\\) from axis \\(A\\) and \\(d_B\\) from the center of mass, the student uses the relation \\(I_B = I_A + M d^2\\). This calculation yields an incorrect value for \\(I_B\\). Which of the following best explains why the student’s calculation is incorrect and how the correct value of \\(I_B\\) must be found?"
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url: "https://nerd-notes.com/ubq/124229/"
date_modified: "2026-09-28T14:04:19+00:00"
---

# A student calculates the rotational inertia \(I_A\) of a rigid body of mass \(M\) about an axis \(A\) located a distance \(d_A\) from the center of mass. To determine the rotational inertia \(I_B\) about a parallel axis \(B\) located a distance \(d\) from axis \(A\) and \(d_B\) from the center of mass, the student uses the relation \(I_B = I_A + M d^2\). This calculation yields an incorrect value for \(I_B\). Which of the following best explains why the student’s calculation is incorrect and how the correct value of \(I_B\) must be found?

A student calculates the rotational inertia \(I_A\) of a rigid body of mass \(M\) about an axis \(A\) located a distance \(d_A\) from the center of mass. To determine the rotational inertia \(I_B\) about a parallel axis \(B\) located a distance \(d\) from axis \(A\) and \(d_B\) from the center of mass, the student uses the relation \(I_B = I_A + M d^2\). This calculation yields an incorrect value for \(I_B\). Which of the following best explains why the student's calculation is incorrect and how the correct value of \(I_B\) must be found?

![An irregularly shaped two-dimensional enclosed outline representing a rigid body of mass \(M\). Inside the outline, a small solid dot labeled \(\text{CM}\) marks the center of mass. A dashed vertical line passes through \(\text{CM}\). To the left of \(\text{CM}\), a small solid dot labeled \(A\) marks axis \(A\), with a parallel vertical dashed line passing through it. A horizontal double-headed arrow between axis \(A\) and \(\text{CM}\) is labeled \(d_A\). To the right of \(\text{CM}\), a small solid dot labeled \(B\) marks axis \(B\), with a parallel vertical dashed line passing through it. A horizontal double-headed arrow between \(\text{CM}\) and axis \(B\) is labeled \(d_B\). A horizontal double-headed arrow spanning directly from axis \(A\) to axis \(B\) is labeled \(d\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604259-2zI2ob.jpg)

- **A.** The linear separation \(d\) fails to account for mass distribution perpendicular to the plane of rotation, so direct scalar addition is invalid; to find \(I_B\), one must use the perpendicular axis theorem to sum the planar moments of inertia about the orthogonal axes passing through axis \(A\).
- **B.** The rotational inertia about an off-center axis is greater than about any axis closer to the geometric edge, so the distance term should be subtracted rather than added; to find \(I_B\), one must calculate \(I_B = I_A - M d^2\) directly without referencing the center of mass.
- **C.** The integral cross-term involving position vanishes only when coordinates are defined relative to the center of mass, so the theorem holds only between an axis and the center of mass; to find \(I_B\), one must first calculate \(I_{\text{cm}} = I_A - M d_A^2\) and then evaluate \(I_B = I_{\text{cm}} + M d_B^2\).
- **D.** The parallel axis theorem is valid only for symmetric objects rotating about a principal axis of inertia where angular momentum is parallel to angular velocity; to find \(I_B\), one must determine the net torque required to displace the axis of rotation dynamically.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124229/*