parallel axis theorem

Question

I_cm of a disc is (1/2)MR². Moment of inertia about tangent in plane of disc?

RankGuru · Accepted Answer

1. **Identify Axis**: The question asks about an axis in the **plane** of the disc that is tangent to the edge.
2. **Step 1: Moment about Diameter ($I_d$)**: 
   - By the Perpendicular Axis Theorem, $I_{center-perpendicular} = I_{dia,x} + I_{dia,y}$.
   - $\frac{1}{2} MR^2 = 2 I_d \implies I_d = \frac{1}{4} MR^2$.
3. **Step 2: Apply Parallel Axis Theorem**: The tangent in the plane is parallel to a diameter and is at distance $R$ from the center.
   - $I_{tangent} = I_{cm,parallel} + M d^2$
   - $I_{tangent} = \frac{1}{4} MR^2 + M R^2$
4. **Calculation**: $I_{tangent} = (1/4 + 1) MR^2 = \frac{5}{4} MR^2$.
5. **Conclusion**: The moment of inertia is **$(5/4)MR^2$**.

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