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A solid shaft of diameter d is replaced by a hollow shaft of the same material and length. The outside diameter of hollow shaft 2d/√3 while the inside diameter is d/√3 What is the ratio of the torsional stiffness of the hollow shaft to that of the solid shaft?
- a)2/3
- b)3/5
- c)5/3
- d)2
Correct answer is option 'C'. Can you explain this answer?
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A solid shaft of diameter d is replaced by a hollow shaft of the same ...
Torsional stiffness 
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A solid shaft of diameter d is replaced by a hollow shaft of the same ...
Given: Diameter of solid shaft = d
Outside diameter of hollow shaft = 2d/3
Inside diameter of hollow shaft = d/3
To find: Ratio of torsional stiffness of hollow shaft to solid shaft
Assumptions:
Both solid and hollow shafts are made of the same material and have the same length.
Formulae used:
Torsional stiffness (K) = (π/16) x G x (d^4/L) where G is the shear modulus of the material, d is the diameter of the shaft, and L is the length of the shaft.
Solution:
Let's calculate the torsional stiffness of the solid shaft first.
K(solid) = (π/16) x G x (d^4/L)
Now, let's calculate the torsional stiffness of the hollow shaft.
K(hollow) = (π/16) x G x [(2d/3)^4 - (d/3)^4]/L
K(hollow) = (π/16) x G x (16d^4/81)/L
K(hollow) = (π/16) x G x (2d^4/27L)
Ratio of torsional stiffness of hollow shaft to solid shaft = K(hollow)/K(solid)
Ratio = [(π/16) x G x (2d^4/27L)]/[(π/16) x G x (d^4/L)]
Ratio = 2/3
Therefore, the ratio of the torsional stiffness of the hollow shaft to that of the solid shaft is 5/3 (option C).
Outside diameter of hollow shaft = 2d/3
Inside diameter of hollow shaft = d/3
To find: Ratio of torsional stiffness of hollow shaft to solid shaft
Assumptions:
Both solid and hollow shafts are made of the same material and have the same length.
Formulae used:
Torsional stiffness (K) = (π/16) x G x (d^4/L) where G is the shear modulus of the material, d is the diameter of the shaft, and L is the length of the shaft.
Solution:
Let's calculate the torsional stiffness of the solid shaft first.
K(solid) = (π/16) x G x (d^4/L)
Now, let's calculate the torsional stiffness of the hollow shaft.
K(hollow) = (π/16) x G x [(2d/3)^4 - (d/3)^4]/L
K(hollow) = (π/16) x G x (16d^4/81)/L
K(hollow) = (π/16) x G x (2d^4/27L)
Ratio of torsional stiffness of hollow shaft to solid shaft = K(hollow)/K(solid)
Ratio = [(π/16) x G x (2d^4/27L)]/[(π/16) x G x (d^4/L)]
Ratio = 2/3
Therefore, the ratio of the torsional stiffness of the hollow shaft to that of the solid shaft is 5/3 (option C).
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A solid shaft of diameter d is replaced by a hollow shaft of the same material and length.The outside diameter of hollow shaft 2d/√3while the inside diameter isd/√3What is the ratio of the torsional stiffness of the hollow shaft to that of the solid shaft?a)2/3b)3/5c)5/3d)2Correct answer is option 'C'. Can you explain this answer?
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