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On page-3 of these MIT OCW notes, it is said that if we want to find the stresses when the square element is rotated by $\theta$ , then we should rotate diamter of mohr circle by $2 \theta$ and read of the numbers from the ends of diameter.

Is there any intuitive reason as to why we have to use twice the angle?

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One intuitive reason is that if you want to rotate by 90 degrees then $\sigma_x$ would be equal to $\sigma_y$, and $\sigma_y\rightarrow \sigma_x$.

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In order for that to happen you'd have to travel 180 degrees. This is better evidenced for principal stresses ($\sigma_1, \sigma2$, however, it can be seen and in the generic stress state

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A stress component is the quotient of a force resolved along a particular axis by the area of a surface perpendicular to a particular axis (in the same set of mutually perpendicular axes), on which that force is applied. (If the two "particular axes" are the same axis, that's a longitudinal component; if they're different axes within the same set of mutually perpendicular axes, that's a shear component.)

When one switches to resolving along a different set of mutually perpendicular axes, rotated by $\theta$ from the original set, one gets one factor of $\cos\left(\theta\right)$ in the stress component from projecting the force onto the new set of axes, and another factor of $\cos\left(\theta\right)$ in the stress component from projecting the surface area onto the new set of axes, making a factor of $\cos^2\left(\theta\right)$ altogether. By a well-known trigonometric identity, that factor is the same as $\left(1/2\right)+\left(1/2\right)\cos\left(2\theta\right)$. The first $(1/2)$ is why the centre of Mohr's circle is offset from the origin of a longitudinal stress-shear stress graph by $(1/2)$ of the sum of principal stresses; the second $(1/2)$ is why the radius of Mohr's circle is $(1/2)$ of the difference between the principal stresses; and the $\left(2\theta\right)$ is why one turns $\left(2\theta\right)$ around Mohr's circle from the longitudinal stress axis of the graph.

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