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Two hollow shafts of the same material have the same length and outside diameter. Shaft 1 has internal diameter equal to one-third of the outer diameter and shaft 2 has internal diameter equal to half of the outer diameter. If both the shafts are subjected to the same torque, the ratio of their twists θ1/θ2 will be equal to:
- a)16/81
- b)8/27
- c)19/27
- d)243/256
Correct answer is option 'D'. Can you explain this answer?
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The twist in a shaft is given by the equation:
θ = (T/J) * L
where θ is the twist angle, T is the torque applied, J is the polar moment of inertia of the shaft, and L is the length of the shaft.
The polar moment of inertia of a hollow shaft is given by:
J = (π/2) * (D^4 - d^4)
where D is the outer diameter and d is the inner diameter.
For shaft 1, D = d + 2d/3 = 5d/3
For shaft 2, D = d + d/2 = 3d/2
Since both shafts have the same outer diameter and length, we can assume that they have the same material and hence the same density. Therefore, the mass of the shafts will be proportional to their volumes, which are given by:
V1 = (π/4) * (D^2 - d^2) * L = (π/4) * (25d^2/9 - d^2) * L = (8π/9) * d^2 * L
V2 = (π/4) * (D^2 - d^2) * L = (π/4) * (9d^2/4 - d^2) * L = (5π/16) * d^2 * L
The ratio of their volumes is therefore:
V1/V2 = (8π/9) * d^2 * L / (5π/16) * d^2 * L = (512/225)
Since the mass and material are the same for both shafts, we can assume that their polar moment of inertia is proportional to their volumes. Therefore, we can write:
J1/J2 = V1/V2 = 512/225
Now, let us consider the twist angle for each shaft. Since the torque is the same for both shafts, we can write:
θ1/θ2 = J1/J2 * L/L = 512/225
Therefore, the ratio of their twists is 512/225.
θ = (T/J) * L
where θ is the twist angle, T is the torque applied, J is the polar moment of inertia of the shaft, and L is the length of the shaft.
The polar moment of inertia of a hollow shaft is given by:
J = (π/2) * (D^4 - d^4)
where D is the outer diameter and d is the inner diameter.
For shaft 1, D = d + 2d/3 = 5d/3
For shaft 2, D = d + d/2 = 3d/2
Since both shafts have the same outer diameter and length, we can assume that they have the same material and hence the same density. Therefore, the mass of the shafts will be proportional to their volumes, which are given by:
V1 = (π/4) * (D^2 - d^2) * L = (π/4) * (25d^2/9 - d^2) * L = (8π/9) * d^2 * L
V2 = (π/4) * (D^2 - d^2) * L = (π/4) * (9d^2/4 - d^2) * L = (5π/16) * d^2 * L
The ratio of their volumes is therefore:
V1/V2 = (8π/9) * d^2 * L / (5π/16) * d^2 * L = (512/225)
Since the mass and material are the same for both shafts, we can assume that their polar moment of inertia is proportional to their volumes. Therefore, we can write:
J1/J2 = V1/V2 = 512/225
Now, let us consider the twist angle for each shaft. Since the torque is the same for both shafts, we can write:
θ1/θ2 = J1/J2 * L/L = 512/225
Therefore, the ratio of their twists is 512/225.
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Two hollow shafts of the same material have the same length and outside diameter. Shaft 1 has internal diameter equal to one-third of the outer diameter and shaft 2 has internal diameter equal to half of the outer diameter. If both the shafts are subjected to the same torque, the ratio of their twists θ1/θ2 will be equal to:a)16/81b)8/27c)19/27d)243/256Correct answer is option 'D'. Can you explain this answer?
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Two hollow shafts of the same material have the same length and outside diameter. Shaft 1 has internal diameter equal to one-third of the outer diameter and shaft 2 has internal diameter equal to half of the outer diameter. If both the shafts are subjected to the same torque, the ratio of their twists θ1/θ2 will be equal to:a)16/81b)8/27c)19/27d)243/256Correct answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Two hollow shafts of the same material have the same length and outside diameter. Shaft 1 has internal diameter equal to one-third of the outer diameter and shaft 2 has internal diameter equal to half of the outer diameter. If both the shafts are subjected to the same torque, the ratio of their twists θ1/θ2 will be equal to:a)16/81b)8/27c)19/27d)243/256Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two hollow shafts of the same material have the same length and outside diameter. Shaft 1 has internal diameter equal to one-third of the outer diameter and shaft 2 has internal diameter equal to half of the outer diameter. If both the shafts are subjected to the same torque, the ratio of their twists θ1/θ2 will be equal to:a)16/81b)8/27c)19/27d)243/256Correct answer is option 'D'. Can you explain this answer?.
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