Let's solve ** 1**-by-step: **Find the external (outer) diameter in terms of the internal (inner) diameter.** --- ## **Step 1: Given Data** - Power, \( P = 600 \) kW \( = 600,000 \) W - Rotational speed, \( N = 200 \) rpm - Maximum shear stress, \( \tau_{max} = 60 \) N/mm\(^2\) \( = 60 \times 10^6 \) N/m\(^2\) - Let: - Internal diameter = \( d_i \) - External diameter = \( d_o \) --- ## **Step 2: Find the Torque (T)** Power transmitted by a shaft: \[ P = 2\pi N T / 60 \] \[ T = \frac{P \times 60}{2\pi N} \] Substitute values: \[ T = \frac{600,000 \times 60}{2\pi \times 200} \] \[ T = \frac{36,000,000}{1256.64} \] \[ T \approx 28,652.6 \text{ Nm} \] --- ## **Step 3: Maximum Shear Stress Formula for Hollow Shaft** The maximum shear stress in a hollow shaft: \[ \tau_{max} = \frac{T \cdot r_o}{J} \] Where: - \( r_o = \) outer radius = \( \frac{d_o}{2} \) - \( J = \) polar moment of inertia for a hollow shaft: \[ J = \frac{\pi}{32} (d_o^4 - d_i^4) \] So, \[ \tau_{max} = \frac{T \cdot \frac{d_o}{2}}{\frac{\pi}{32}(d_o^4 - d_i^4)} \] \[ \tau_{max} = \frac{16 T d_o}{\pi (d_o^4 - d_i^4)} \] --- ## **Step 4: Rearrange to Express \( d_o \) in terms of \( d_i \)** Rearrange the above equation: \[ \tau_{max} = \frac{16 T d_o}{\pi (d_o^4 - d_i^4)} \] \[ \tau_{max} \cdot \pi (d_o^4 - d_i^4) = 16 T d_o \] \[ \tau_{max} \cdot \pi (d_o^4 - d_i^4) - 16 T d_o = \] \[ d_o^4 - d_i^4 = \frac{16T}{\tau_{max} \pi} d_o \] \[ d_o^4 - \frac{16T}{\tau_{max} \pi} d_o - d_i^4 = \] This is the cubic equation in terms of \( d_o \): --- ## **Step 5: Final Answer** **The external (outer) diameter \( d_o \) in terms of internal (inner) diameter \( d_i \):** \[ \boxed{ d_o^4 - \frac{16T}{\tau_{max} \pi} d_o - d_i^4 = } \] Where: - \( T = \) torque transmitted (calculated above) - \( \tau_{max} = \) maximum shear stress --- ### **Summary Table** | Symbol | Meaning | |---------------|----------------------------------| | \( d_o \) | Outer diameter | | \( d_i \) | Inner diameter | | \( T \) | Torque (\(\approx 28,652.6\) Nm) | | \( \tau_{max} \) | Max shear stress (60 N/mm\(^2\)) | --- If you know the ratio \( d_i/d_o \) or want to solve for \( d_o \) given \( d_i \), substitute all values and solve the quartic equation. --- **Let me know if you want the numerical value for \( d_o \) for a specific \( d_i \), or if you want the next steps for the angle of twist!**