https://file.aiquickdraw.com/mathbot/file/1725612755552871m3atj.jpg

Let's work through the problem outlined in the image step-by-step.

### Given Information:

- **Torques:** 200 Nm at A and 1000 Nm at B.
- **Hole in portion CD:** Diameter of 40 mm.
- **Material:** Shaft is made of steel with \( G = 77 \, \text{GPa} \).
- **Shaft details:** We're solving for stresses, torques, and angles of twist in various sections of the shaft.

#### Required:

1. **Stress in section AB due to the applied 50 kN force** (This seems like it refers to the shear stress induced by the torque, not directly the force).
2. **Stress in sections AB, BC, and CD due to applied torques**.
3. **Angle of twist between AB, BC, and CD in degrees**.
4. **Angle of twist in degrees between A and D**.

---

### Key Formulas:
1. **Polar Moment of Inertia** \( J \) for solid and hollow circular sections:
   \[
   J_{\text{solid}} = \frac{\pi d^4}{32}, \quad J_{\text{hollow}} = \frac{\pi (d_{\text{outer}}^4 - d_{\text{inner}}^4)}{32}
   \]

2. **Shear stress due to torque**:
   \[
   \tau = \frac{T r}{J}
   \]
   where \( T \) is torque, \( r \) is the radius, and \( J \) is the polar moment of inertia.

3. **Angle of twist** \( \theta \):
   \[
   \theta = \frac{T L}{G J}
   \]
   where \( T \) is torque, \( L \) is length, \( G \) is shear modulus, and \( J \) is the polar moment of inertia.

---

### Let me compute each of the required parts for you. I'll start with the polar moment of inertia for each section of the shaft, and then we can move on to stresses and angles of twist.Here are the results for the angles of twist:

1. **Angle of twist between section AB**: \( 0.18^\circ \)
2. **Angle of twist between section BC**: \( 1.78^\circ \)
3. **Angle of twist between section CD**: \( 1.48^\circ \)
4. **Total angle of twist between A and D**: \( 3.43^\circ \)

Next, we can calculate the stresses in each section based on the torques. Would you like to proceed with that or explore any other detail first?

### Related Questions:
1. What is the shear stress due to torque in section AB?
2. How does the stress differ between the solid and hollow sections?
3. Can you derive the formula for stress based on the applied torque?
4. How would the results change if the shaft had a different material with a lower shear modulus \( G \)?
5. What would happen if the applied torque at point A was increased?

### Tip:
Always ensure to use consistent units (especially meters, seconds, and Newtons) to avoid errors during calculations!

The horizontal shaft AD is attached to a fixed base at D and is subjected to the torques of 200 Nm at A and 1000 Nm at B. Determine the stress in various sections of the shaft and the angle of twist between different points.

This problem involves calculating shear stress and angle of twist in a shaft subjected to multiple torques. The shaft has different sections, including a hollow portion. It requires the use of torsion mechanics and material properties like shear modulus to find the final deformation.

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