A solid circular shaft with a diameter of 50 mm is subjected to a torque of 100 Nm . What is the maximum shear stress?

To find the maximum shear stress in a solid circular shaft, you can use the formula: \[ \tau_{max} = \frac{T \cdot r}{J} \] where \( T \) is the torque, \( r \) is the radius, and \( J \) is the polar moment of inertia. 1. **Calculate the radius**: The diameter is 50 mm, so the radius \( r = \frac{50}{2} = 25 \) mm or 0.025 m. 2. **Calculate the polar moment of inertia \( J \)**: For a solid circular shaft, \( J = \frac{\pi r^4}{2} = \frac{\pi (0.025)^4}{2} \approx 4.9 \times 10^{-9} \, \text{m}^4\). 3. **Substitute the values**: \[ \tau_{max} = \frac{100 \times 0.025}{4.9 \times 10^{-9}} \approx 510,204.08 \, \text{Pa} \, \text{or} \, 510.2 \, \text{kPa}. \] So, the maximum shear stress in the shaft is approximately 510.2 kPa.