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Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:
- a)π/8(D4−d4)
- b)π/32(D4−d4)
- c)π/64(D4−d4)
- d)π/12(D4−d4)
Correct answer is option 'B'. Can you explain this answer?
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Polar moment of inertia of a hollow shaft having D and d as outer and...
Polar moment of inertia of Solid cylinder
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IZ=J=πD4/32
Polar moment of inertia of Hollow Cylinder:
IZ = J = π(D4−d4)/32
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Polar moment of inertia of a hollow shaft having D and d as outer and...
Understanding Polar Moment of Inertia
The polar moment of inertia (J) is a measure of an object's resistance to torsion. For a hollow shaft, it depends on the outer diameter (D) and inner diameter (d).
Formula Derivation
The polar moment of inertia for a hollow shaft can be derived from the standard formula for the moment of inertia:
- The moment of inertia for a solid cylinder is given by:
- \( J = \frac{π}{2}R^4 \)
- For a hollow cylinder, the moment of inertia is calculated by subtracting the inner cylinder's moment of inertia from the outer cylinder's moment of inertia:
- \( J = J_{outer} - J_{inner} \)
- Where:
- \( J_{outer} = \frac{π}{2} \left( \frac{D}{2} \right)^4 = \frac{πD^4}{32} \)
- \( J_{inner} = \frac{π}{2} \left( \frac{d}{2} \right)^4 = \frac{πd^4}{32} \)
- Thus, the polar moment of inertia becomes:
- \( J = \frac{πD^4}{32} - \frac{πd^4}{32} \)
- Simplifying this yields:
- \( J = \frac{π}{32}(D^4 - d^4) \)
Final Result
Hence, the polar moment of inertia for a hollow shaft with outer diameter D and inner diameter d is:
- **Correct Answer:** \( \frac{π}{32}(D^4 - d^4) \)
This confirms that option 'B' is indeed the correct answer. Understanding this derivation is crucial for applications in mechanical and civil engineering, particularly in analyzing torsional stresses in hollow shafts.
The polar moment of inertia (J) is a measure of an object's resistance to torsion. For a hollow shaft, it depends on the outer diameter (D) and inner diameter (d).
Formula Derivation
The polar moment of inertia for a hollow shaft can be derived from the standard formula for the moment of inertia:
- The moment of inertia for a solid cylinder is given by:
- \( J = \frac{π}{2}R^4 \)
- For a hollow cylinder, the moment of inertia is calculated by subtracting the inner cylinder's moment of inertia from the outer cylinder's moment of inertia:
- \( J = J_{outer} - J_{inner} \)
- Where:
- \( J_{outer} = \frac{π}{2} \left( \frac{D}{2} \right)^4 = \frac{πD^4}{32} \)
- \( J_{inner} = \frac{π}{2} \left( \frac{d}{2} \right)^4 = \frac{πd^4}{32} \)
- Thus, the polar moment of inertia becomes:
- \( J = \frac{πD^4}{32} - \frac{πd^4}{32} \)
- Simplifying this yields:
- \( J = \frac{π}{32}(D^4 - d^4) \)
Final Result
Hence, the polar moment of inertia for a hollow shaft with outer diameter D and inner diameter d is:
- **Correct Answer:** \( \frac{π}{32}(D^4 - d^4) \)
This confirms that option 'B' is indeed the correct answer. Understanding this derivation is crucial for applications in mechanical and civil engineering, particularly in analyzing torsional stresses in hollow shafts.
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Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:a)π/8(D4−d4)b)π/32(D4−d4)c)π/64(D4−d4)d)π/12(D4−d4)Correct answer is option 'B'. Can you explain this answer?
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Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:a)π/8(D4−d4)b)π/32(D4−d4)c)π/64(D4−d4)d)π/12(D4−d4)Correct answer is option 'B'. Can you explain this answer? for Railways 2024 is part of Railways preparation. The Question and answers have been prepared according to the Railways exam syllabus. Information about Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:a)π/8(D4−d4)b)π/32(D4−d4)c)π/64(D4−d4)d)π/12(D4−d4)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Railways 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:a)π/8(D4−d4)b)π/32(D4−d4)c)π/64(D4−d4)d)π/12(D4−d4)Correct answer is option 'B'. Can you explain this answer?.
Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:a)π/8(D4−d4)b)π/32(D4−d4)c)π/64(D4−d4)d)π/12(D4−d4)Correct answer is option 'B'. Can you explain this answer? for Railways 2024 is part of Railways preparation. The Question and answers have been prepared according to the Railways exam syllabus. Information about Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:a)π/8(D4−d4)b)π/32(D4−d4)c)π/64(D4−d4)d)π/12(D4−d4)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Railways 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Polar moment of inertia of a hollow shaft having D and d as outer and inner diameters, is given by:a)π/8(D4−d4)b)π/32(D4−d4)c)π/64(D4−d4)d)π/12(D4−d4)Correct answer is option 'B'. Can you explain this answer?.
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