The problem appears to be related to determining the stress components in a shaft under a load. It involves both torsional and bending stress analysis. The diagram shows a rotating shaft with dimensions and load distribution, along with bearings at both ends.

Key elements:

• Shaft rotation speed: 1725 rpm
• Shaft dimensions:
  • Diameter at different points: 1.50 inches, 1.25 inches
  • Keyway dimensions: r = 0.063 inches and r = 0.125 inches
  • Lengths: Different segments of 8 inches and 10 inches along the shaft.
• Loading: A point load $P$ at the midpoint of the shaft, which will induce bending moments and potentially torsional moments.

Steps to Solve:

1. Determine Reaction Forces: Using static equilibrium, calculate the reaction forces at the bearings $R_L$ and $R_R$.
2. Draw Shear and Moment Diagrams: This will help visualize how the bending moment varies along the length of the shaft.
3. Calculate Bending Stress: Using the bending moment at the location directly under the load $P$, calculate the bending stress using the formula: $\sigma_b = \frac{M \cdot c}{I}$ where $M$ is the bending moment, $c$ is the distance from the neutral axis, and $I$ is the second moment of area.
4. Calculate Torsional Stress: If there is torsion involved (due to the load or due to the rotation), the torsional stress can be calculated using: $\tau_t = \frac{T \cdot c}{J}$ where $T$ is the torque and $J$ is the polar moment of inertia.
5. Combine Stresses: Use the appropriate theory of failure (such as maximum shear stress theory or von Mises stress) to combine the normal and shear stresses to find the resultant stress at the critical point.

Would you like to proceed with a detailed step-by-step solution for calculating the stress components at a specific point on the shaft?

Here are some related questions to deepen your understanding:

1. What role do the bearings play in distributing the load on the shaft?
2. How would you calculate the second moment of area $I$ for a circular cross-section?
3. What is the significance of the keyway in terms of stress concentration?
4. How does the speed of rotation (1725 rpm) affect the stress analysis?
5. What is the difference between bending stress and torsional stress in a shaft?

Tip: Remember that the presence of a keyway typically introduces stress concentrations, which should be accounted for when determining the maximum stress.