A semicircular lamina has radius R and diameter D. Determine the moment of inertia about the diametrical axis as shown below?
- AπD⁴ / 32
- BπR⁴ / 8Correct
- CπR⁴ / 16
- DπD⁴ / 64
Solution
Concept
MOI of a semicircle about its diameter is half the MOI of a full circle about a diameter.
Formula
Idiameter = πR4/8 = πD4/128
Step-by-step solution
1. MOI of full circle about a diameter = πD4/64 = πR4/4. 2. The semicircle is exactly half of the circle about the same diameter line. 3. So I = (1/2) × πR4/4 = πR4/8. 4. In terms of D: πR4/8 = π(D/2)4/8 = πD4/128.
Explanation
The base diameter of the semicircle lies on the axis shown. The MOI of a semicircle about this diameter is half that of a full circle, so it is πR4/8.
Why the other options are wrong
A: πD4/32 is the polar MOI of a full circle. C: πR4/16 is the MOI of a quarter circle about one straight edge. D: πD4/64 is the MOI of a full circle, not a semicircle.
Shortcut
Semicircle about its diameter = half of the full circle value = πR4/8.
Common mistake
Choosing πD
Correct answer: B) πR⁴ / 8