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A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft is
- a)2.5 cm
- b)3.5 cm
- c)4.5 cm
- d)4.0 cm
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replace...
Now, 
∴ 
= 0.9036d1 = 5 × 0.9036 = 4.518 cm
Most Upvoted Answer
A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replace...
To find the inner diameter of the tubular shaft, we need to compare the torsional rigidity of the two shafts.
1. Calculate the torsional rigidity of the solid aluminium shaft:
Torsional rigidity (J) is given by the formula J = (π/32) * (d^4), where d is the diameter of the shaft.
For the solid aluminium shaft, d = 5 cm = 0.05 m.
J_aluminium = (π/32) * (0.05^4) = 6.25e-8 m^4.
2. Calculate the torsional rigidity of the tubular steel shaft:
Since the outside diameter of the tubular steel shaft is the same as the solid aluminium shaft, we can use the same formula for the outside diameter.
For the tubular steel shaft, d = 5 cm = 0.05 m.
J_steel_outside = (π/32) * (0.05^4) = 6.25e-8 m^4.
The inside diameter of the tubular steel shaft is denoted by d' (to be determined).
3. Calculate the inner torsional rigidity of the tubular steel shaft:
The inner torsional rigidity (J') of the tubular steel shaft can be obtained by subtracting the torsional rigidity of the hollow portion from the torsional rigidity of the solid portion.
J' = J_steel_outside - J_steel_inside.
4. Use the given information to relate the modulus of rigidity of steel to that of aluminium:
The modulus of rigidity (G) is related to the torsional rigidity (J) by the formula G = (J * L) / (θ * T), where L is the length of the shaft, θ is the angle of twist, and T is the torsional moment.
Since we want the two shafts to have the same angle of twist per unit torsional moment, we can write the equation as G_aluminium = G_steel, where G_aluminium = (J_aluminium * L) / T and G_steel = (J_steel_outside * L) / T.
5. Set up the equation to find the inner torsional rigidity of the tubular steel shaft:
G_aluminium = G_steel
(J_aluminium * L) / T = (J_steel_outside * L) / T - (J' * L) / T.
6. Substitute the values and solve for J':
(J_aluminium * L) = (J_steel_outside * L) - (J' * L)
(J_aluminium * L) = (J_steel_outside * L) - (J_steel_outside - J_steel_inside) * L
(J_aluminium * L) = (J_steel_outside * L) - (J_steel_outside * L) + (J_steel_inside * L)
(J_aluminium * L) = (J_steel_inside * L)
J_aluminium = J_steel_inside.
7. Substitute the values of J_aluminium and J_steel_inside to find the inner diameter of the tubular shaft:
J_aluminium = (π/32) * (0.05^4) = J_steel_inside
(π/32) * (0.05^4) = (π
1. Calculate the torsional rigidity of the solid aluminium shaft:
Torsional rigidity (J) is given by the formula J = (π/32) * (d^4), where d is the diameter of the shaft.
For the solid aluminium shaft, d = 5 cm = 0.05 m.
J_aluminium = (π/32) * (0.05^4) = 6.25e-8 m^4.
2. Calculate the torsional rigidity of the tubular steel shaft:
Since the outside diameter of the tubular steel shaft is the same as the solid aluminium shaft, we can use the same formula for the outside diameter.
For the tubular steel shaft, d = 5 cm = 0.05 m.
J_steel_outside = (π/32) * (0.05^4) = 6.25e-8 m^4.
The inside diameter of the tubular steel shaft is denoted by d' (to be determined).
3. Calculate the inner torsional rigidity of the tubular steel shaft:
The inner torsional rigidity (J') of the tubular steel shaft can be obtained by subtracting the torsional rigidity of the hollow portion from the torsional rigidity of the solid portion.
J' = J_steel_outside - J_steel_inside.
4. Use the given information to relate the modulus of rigidity of steel to that of aluminium:
The modulus of rigidity (G) is related to the torsional rigidity (J) by the formula G = (J * L) / (θ * T), where L is the length of the shaft, θ is the angle of twist, and T is the torsional moment.
Since we want the two shafts to have the same angle of twist per unit torsional moment, we can write the equation as G_aluminium = G_steel, where G_aluminium = (J_aluminium * L) / T and G_steel = (J_steel_outside * L) / T.
5. Set up the equation to find the inner torsional rigidity of the tubular steel shaft:
G_aluminium = G_steel
(J_aluminium * L) / T = (J_steel_outside * L) / T - (J' * L) / T.
6. Substitute the values and solve for J':
(J_aluminium * L) = (J_steel_outside * L) - (J' * L)
(J_aluminium * L) = (J_steel_outside * L) - (J_steel_outside - J_steel_inside) * L
(J_aluminium * L) = (J_steel_outside * L) - (J_steel_outside * L) + (J_steel_inside * L)
(J_aluminium * L) = (J_steel_inside * L)
J_aluminium = J_steel_inside.
7. Substitute the values of J_aluminium and J_steel_inside to find the inner diameter of the tubular shaft:
J_aluminium = (π/32) * (0.05^4) = J_steel_inside
(π/32) * (0.05^4) = (π
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A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft isa)2.5 cmb)3.5 cmc)4.5 cmd)4.0 cmCorrect answer is option 'C'. Can you explain this answer?
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A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft isa)2.5 cmb)3.5 cmc)4.5 cmd)4.0 cmCorrect answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft isa)2.5 cmb)3.5 cmc)4.5 cmd)4.0 cmCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft isa)2.5 cmb)3.5 cmc)4.5 cmd)4.0 cmCorrect answer is option 'C'. Can you explain this answer?.
A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft isa)2.5 cmb)3.5 cmc)4.5 cmd)4.0 cmCorrect answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft isa)2.5 cmb)3.5 cmc)4.5 cmd)4.0 cmCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A solid aluminium shaft, 5 cm diameter and 1 m long, is to be replaced by a tubular steel shaft of the same length and the same outside diameter (i.e. 5 cm), such that each of the two shaft could have the same angle of twist per unit torsional moment over the total length. Modulus of rigidity of steel is three times that of aluminium. The inner diameter of the tubular shaft isa)2.5 cmb)3.5 cmc)4.5 cmd)4.0 cmCorrect answer is option 'C'. Can you explain this answer?.
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