Question
Download Solution PDFDetermine the torsional rigidity of a hollow shaft of 200 mm external diameter and 150 mm internal diameter. Consider G = 90 GPa.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
We use the polar moment of inertia and shear modulus to determine the torsional rigidity of a hollow shaft.
Given:
- External diameter, \( D = 200 \, \text{mm} \)
- Internal diameter, \( d = 150 \, \text{mm} \)
- Shear modulus, \( G = 90 \, \text{GPa} = 90 \times 10^3 \, \text{N/mm}^2 \)
Step 1: Calculate Polar Moment of Inertia (J)
\( J = \frac{\pi}{32} \left( D^4 - d^4 \right) \)
\( J = \frac{\pi}{32} \left( (200)^4 - (150)^4 \right) \)
\( J = \frac{\pi}{32} \left( 1.6 \times 10^9 - 5.0625 \times 10^8 \right) \)
\( J = \frac{\pi}{32} \left( 1.09375 \times 10^9 \right) \)
\( J \approx 1.073 \times 10^8 \, \text{mm}^4 \)
Step 2: Compute Torsional Rigidity (GJ)
\( GJ = G \times J \)
\( GJ = 90 \times 10^3 \, \text{N/mm}^2 \times 1.073 \times 10^8 \, \text{mm}^4 \)
\( GJ \approx 9.66 \times 10^{12} \, \text{N-mm}^2 \)
Last updated on May 28, 2025
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