To solve the problem using the provided Mohr's circle, we need to determine the angle of inclination of the principal plane, denoted as $\theta_p$, based on the Mohr's circle diagram.

Key Observations from the Diagram:

1. Mohr's Circle Basics:

   • The center of the Mohr's circle represents the average normal stress, $\sigma_{\text{avg}}$.
   • The radius of the circle represents the maximum shear stress.
   • The x-axis typically represents normal stress ($\sigma$), and the y-axis represents shear stress ($\tau$).

2. Angle in Mohr’s Circle:

   • In Mohr's circle, the angle $2\theta_p$ (where $\theta_p$ is the angle of the principal plane in the actual material) is the angle between the line connecting the point on the circle corresponding to the given stress state and the horizontal axis (line OX in the figure).

Steps to Determine the Angle of Inclination:

1. Identify $2\theta_p$ from the Circle:

   • The angle between the given stress point (where the stress state lies on Mohr's circle) and the principal stress direction on the circle is $2\theta_p$. This is typically measured in the circle diagram from the given stress state to the point where the principal stress lies.

2. Determine the Angle of Inclination, $\theta_p$:

   • The angle in the material (physical problem) is $\theta_p$, which is half of the angle $2\theta_p$ observed in the Mohr's circle.

Solution:

• From the diagram, the angle $2\theta_p$ appears to be the angle labeled in the Mohr’s circle.
• The inclination angle of the principal plane $\theta_p$ is half of this angle.

Without specific numerical values from the diagram, the general approach to solving would be:

$\theta_p = \frac{2\theta_p}{2}$

If the diagram shows $2\theta_p$, the required $\theta_p$ can be computed directly by halving the angle $2\theta_p$ that is illustrated.

Please provide any angle measurements from the diagram if present, or clarify if there's specific data to be extracted for a more precise calculation.

Would you like more details on Mohr’s circle, or do you have any questions?

Related Questions:

1. What is the purpose of using Mohr's circle in stress analysis?
2. How do you determine the maximum shear stress using Mohr's circle?
3. How does the orientation of the principal plane relate to the principal stresses?
4. What role does the center of Mohr’s circle play in understanding the stress state?
5. How can Mohr’s circle be used to determine the angle between two stress planes?

Tip: Always remember that in Mohr’s circle, the angle $2\theta_p$ is the angle corresponding to the actual physical angle $\theta_p$, making the circle a powerful tool for visualizing the stress transformations.