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Chapter 6: Problem 25

By considering the rotational equilibrium of a fluid mass element, show that \(\tau_{x y}=\tau_{y x}\).

Short Answer

Expert verified
The shear stress components \(\tau_{xy}\) and \(\tau_{yx}\) are equal, i.e., \(\tau_{xy} = \tau_{yx}\) due to the requirement of the rotational equilibrium of a fluid mass element.

Step by step solution

01

Understanding Rotational Equilibrium

The rotational equilibrium implies that the sum of moments about any point inside the fluid must be zero. For rotational equilibrium to be satisfied, the net moment acting on the fluid element must be zero.
02

Calculate Moments Applied by Shear Stresses

The shear stresses \(\tau_{xy}\) and \(\tau_{yx}\) apply opposing moments about the center of the fluid mass element. If the area of each face is A and the length of each side is l, then the moment applied by \(\tau_{xy}\) is \(\tau_{xy} \cdot A \cdot \frac{l}{2}\) and the moment applied by \(\tau_{yx}\) is \(\tau_{yx} \cdot A \cdot \frac{l}{2}\). These moments have opposite directions.
03

Equating the Moments

For the fluid element to be in rotational equilibrium, the moments applied by these two stresses must be balanced i.e., must be equal in magnitude. Hence, we have \(\tau_{xy} \cdot A \cdot \frac{l}{2} = \tau_{yx} \cdot A \cdot \frac{l}{2}\)
04

Solve the Equations

We can cancel out the area A and side length \(l/2\) from both sides, leading to \(\tau_{xy} = \tau_{yx}\)

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Most popular questions from this chapter

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