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The value of shear stress which is induced in the shaft due to the applied couple varies
- a)from maximum at the center to zero at circumference
- b)from zero at the center to maximum at the circumference
- c)from maximum at the center to minimum at the circumference
- d)from minimum at the center to maximum at the circumference
Correct answer is option 'B'. Can you explain this answer?
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Answer:
The correct option is b) from zero at the center to maximum at the circumference.
Explanation:
When a couple is applied to a shaft, it tends to twist the shaft. This twisting effect is resisted by the shear stress induced in the shaft. The distribution of shear stress along the cross-section of the shaft is not uniform. It varies from zero at the center to maximum at the circumference.
The reason behind this variation can be explained by considering a small element of the shaft as shown in the figure below.
- Let us consider a small element of the shaft of radius r and thickness dr at a distance x from the center.
- Due to the applied couple, a shear force is developed in the element which can be given by F = T*r/x, where T is the applied torque.
- The shear stress induced in the element can be given by τ = F/A, where A is the area of the element.
- The area of the element is given by dA = 2πr*dr.
- Therefore, the shear stress can be expressed as τ = T*r/(2πrx*dr).
- From the above expression, it can be seen that the shear stress is directly proportional to the radius r and the torque T, and inversely proportional to the distance x and the thickness dr.
- As x increases, the shear stress decreases, and as r increases, the shear stress increases.
- At the center of the shaft, x = 0, and hence the shear stress is zero.
- At the circumference, x = R, and hence the shear stress is maximum.
Therefore, the correct option is b) from zero at the center to maximum at the circumference.
The correct option is b) from zero at the center to maximum at the circumference.
Explanation:
When a couple is applied to a shaft, it tends to twist the shaft. This twisting effect is resisted by the shear stress induced in the shaft. The distribution of shear stress along the cross-section of the shaft is not uniform. It varies from zero at the center to maximum at the circumference.
The reason behind this variation can be explained by considering a small element of the shaft as shown in the figure below.
- Let us consider a small element of the shaft of radius r and thickness dr at a distance x from the center.
- Due to the applied couple, a shear force is developed in the element which can be given by F = T*r/x, where T is the applied torque.
- The shear stress induced in the element can be given by τ = F/A, where A is the area of the element.
- The area of the element is given by dA = 2πr*dr.
- Therefore, the shear stress can be expressed as τ = T*r/(2πrx*dr).
- From the above expression, it can be seen that the shear stress is directly proportional to the radius r and the torque T, and inversely proportional to the distance x and the thickness dr.
- As x increases, the shear stress decreases, and as r increases, the shear stress increases.
- At the center of the shaft, x = 0, and hence the shear stress is zero.
- At the circumference, x = R, and hence the shear stress is maximum.
Therefore, the correct option is b) from zero at the center to maximum at the circumference.
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The value of shear stress which is induced in the shaft due to the applied couple variesa)from maximum at the center to zero at circumferenceb)from zero at the center to maximum at the circumferencec)from maximum at the center to minimum at the circumferenced)from minimum at the center to maximum at the circumferenceCorrect answer is option 'B'. Can you explain this answer?
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