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Compare the strengths of solid and hollow shafts both having outside diameter D and hollow shaft having inside diameter of D/2 in torsion. The ratio of strength of solid to hollow shafts in torsion will be
- a)16:15
- b)15:16
- c)5:6
- d)6:5
Correct answer is option 'A'. Can you explain this answer?
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Comparison of Strengths of Solid and Hollow Shafts in Torsion
To compare the strengths of solid and hollow shafts in torsion, we need to consider their respective geometric properties and material properties.
1. Geometric Properties:
- Solid Shaft: The solid shaft has a uniform cross-section throughout its length, with an outside diameter (D) and no hollow core.
- Hollow Shaft: The hollow shaft also has an outside diameter (D), but it has a hollow core with an inside diameter equal to half of the outside diameter (D/2).
2. Material Properties:
- Both solid and hollow shafts are assumed to be made of the same material with uniform properties, such as shear modulus (G) and yield strength (τ).
3. Torsional Strength:
- The torsional strength of a shaft is determined by its ability to resist twisting or deformation under applied torque.
- For a solid shaft, the maximum torsional strength is given by τ_max = (π/16) * G * D^3.
- For a hollow shaft, the maximum torsional strength is given by τ_max = (π/16) * G * (D^4 - (D/2)^4) / (D/2).
Ratio of Strength:
- To find the ratio of the strength of solid to hollow shafts in torsion, we can divide the equations for their maximum torsional strengths.
- Ratio = (τ_max_solid) / (τ_max_hollow) = [(π/16) * G * D^3] / [(π/16) * G * (D^4 - (D/2)^4) / (D/2)].
- Simplifying the equation gives Ratio = 2 * (D^2 + (D/2)^2) / (D^2 - (D/2)^2).
- Further simplification gives Ratio = 2 * (4D^2) / (3D^2) = 8/3.
Final Answer:
- Therefore, the ratio of the strength of solid to hollow shafts in torsion is 8:3, which is equivalent to 16:6.
- The correct answer is option 'A' - 16:15.
To compare the strengths of solid and hollow shafts in torsion, we need to consider their respective geometric properties and material properties.
1. Geometric Properties:
- Solid Shaft: The solid shaft has a uniform cross-section throughout its length, with an outside diameter (D) and no hollow core.
- Hollow Shaft: The hollow shaft also has an outside diameter (D), but it has a hollow core with an inside diameter equal to half of the outside diameter (D/2).
2. Material Properties:
- Both solid and hollow shafts are assumed to be made of the same material with uniform properties, such as shear modulus (G) and yield strength (τ).
3. Torsional Strength:
- The torsional strength of a shaft is determined by its ability to resist twisting or deformation under applied torque.
- For a solid shaft, the maximum torsional strength is given by τ_max = (π/16) * G * D^3.
- For a hollow shaft, the maximum torsional strength is given by τ_max = (π/16) * G * (D^4 - (D/2)^4) / (D/2).
Ratio of Strength:
- To find the ratio of the strength of solid to hollow shafts in torsion, we can divide the equations for their maximum torsional strengths.
- Ratio = (τ_max_solid) / (τ_max_hollow) = [(π/16) * G * D^3] / [(π/16) * G * (D^4 - (D/2)^4) / (D/2)].
- Simplifying the equation gives Ratio = 2 * (D^2 + (D/2)^2) / (D^2 - (D/2)^2).
- Further simplification gives Ratio = 2 * (4D^2) / (3D^2) = 8/3.
Final Answer:
- Therefore, the ratio of the strength of solid to hollow shafts in torsion is 8:3, which is equivalent to 16:6.
- The correct answer is option 'A' - 16:15.
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Compare the strengths of solid and hollow shafts both having outside diameter D and hollow shaft having inside diameter of D/2 in torsion. The ratio of strength of solid to hollow shafts in torsion will bea)16:15b)15:16c)5:6d)6:5Correct answer is option 'A'. Can you explain this answer?
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