When can you skip the parallel axis theorem in this setup?

Conditions

• The requested axis is one of the coordinate axes (e.g., the x-axis or y-axis).
• The integral is set up directly about that same axis.
• No separate shifted-axis correction is being applied in the setup.

Reasoning, step by step

1. Read the problem statement carefully to identify the axis about which the moment of inertia is required.
2. Check if the requested axis is a coordinate axis or a centroidal axis.
3. If it is a coordinate axis and the integral is set up directly about it, conclude that the parallel axis theorem is not needed.
4. If it is a centroidal axis or an axis parallel to a coordinate axis, you may need to use the parallel axis theorem to shift the moment of inertia.

Example

The instructor explicitly says, "if it's about one of the coordinate axes you don't have to use the parallel axis theorem because there is no perpendicular distance to the axis." The board text reads "Goal: Find $I_y$ about the y-axis - Centroidal axis," but the instructor treats it as a direct coordinate axis calculation for the setup.

Common misconceptions

• Assuming the parallel axis theorem is always required for any moment of inertia problem.
• Confusing a centroidal axis with a coordinate axis, leading to incorrect application of the theorem.
• Forgetting to check the specific axis requested in the problem statement.

Watch the explanation

Explore next

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• What is the final numerical value of $I_y$ in this worked example?