The state of stress at a point in a component is represented by a Mohr’s circle of radius 100 MPa centered at 200 MPa on the normal stress axis. On a plane passing through the same point, the normal stress is 260 MPa. The magnitude of the shear stress on the same plane at the same point is _______ MPa.

The correct answer is 80.

Step-by-Step Explanation:

Mohr's circle is a graphical representation of the state of stress at a point. The horizontal axis represents the normal stress ($\sigma$), and the vertical axis represents the shear stress ($\tau$).

The equation of a Mohr's circle centered at $({\sigma }_c,0)$ on the normal stress axis with a radius $R$ is given by:
${(\sigma -{\sigma }_c)}^2+{\tau }^2=R^2$

From the given problem, we have the following values:
- Center of the circle on the normal stress axis, ${\sigma }_c=200\text{MPa}$
- Radius of the circle, $R=100\text{MPa}$
- Normal stress on the given plane, $\sigma =260\text{MPa}$

Substituting these values into the equation of the circle:
${(260-200)}^2+{\tau }^2={100}^2$

Simplifying the term inside the parentheses:
${60}^2+{\tau }^2={100}^2$

Rearranging the equation to solve for the magnitude of the shear stress $\tau$:
${\tau }^2={100}^2-{60}^2$
${\tau }^2=10000-3600$
${\tau }^2=6400$
$\tau =\sqrt{6400}=80\text{MPa}$

Therefore, the magnitude of the shear stress on the same plane is 80 MPa.