A clutch connects shaft A to shaft B. Shaft A rotates at 250 rpm. Shaft B starts from rest and reaches full speed in 8 seconds. Calculate the torque needed to accelerate shaft B.

Step 1: Convert given values to SI units.
The external diameter is $D_o = 220 \text{ mm}$, so the external radius is $R_o = \frac{220}{2} = 110 \text{ mm} = 0.110 \text{ m}$.
The internal diameter is $D_i = 110 \text{ mm}$, so the internal radius is $R_i = \frac{110}{2} = 55 \text{ mm} = 0.055 \text{ m}$.
The speed of shaft A is $N_A = 250 \text{ rpm}$.
The time to reach full speed is $t = 8 \text{ s}$.
The coefficient of friction is $\mu = 0.3$.
The moment of inertia of shaft B is $I_B = 30 \text{ kg} \cdot \text{m}^2$.

Step 2: Calculate the angular velocity of shaft A.
The angular velocity $\omega_A$ is given by:

$$\omega_A = \frac{2 \pi N_A}{60}$$ $$\omega_A = \frac{2 \pi (250)}{60} = \frac{500 \pi}{60} = \frac{25 \pi}{3} \text{ rad/s}$$ $$\omega_A \approx 26.1799 \text{ rad/s}$$

Step 3: Calculate the angular acceleration of shaft B.
Shaft B starts from rest ($\omega_{B,initial} = 0$) and reaches the speed of shaft A ($\omega_{B,final} = \omega_A$) in $t = 8$ seconds.

$$\alpha_B = \frac{\omega_{B,final} - \omega_{B,initial}}{t}$$ $$\alpha_B = \frac{\frac{25 \pi}{3} - 0}{8} = \frac{25 \pi}{24} \text{ rad/s}^2$$ $$\alpha_B \approx 3.2725 \text{ rad/s}^2$$

Step 4: Calculate the torque transmitted by the clutch.
The torque $T$ required to accelerate shaft B is given by:

$$T = I_B \alpha_B$$ $$T = (30 \text{ kg} \cdot \text{m}^2) \left(\frac{25 \pi}{24} \text{ rad/s}^2\right)$$ $$T = \frac{30 \times 25 \pi}{24} = \frac{5 \times 25 \pi}{4} = \frac{125 \pi}{4} \text{ N} \cdot \text{m}$$ $$T \approx 98.1748 \text{ N} \cdot \text{m}$$

Step 5: Calculate the mean radius for uniform pressure.
For uniform pressure, the mean radius $R_m$ is given by:

$$R_m = \frac{2}{3} \frac{R_o^3 - R_i^3}{R_o^2 - R_i^2}$$ $$R_m = \frac{2}{3} \frac{(0.110 \text{ m})^3 - (0.055 \text{ m})^3}{(0.110 \text{ m})^2 - (0.055 \text{ m})^2}$$ $$R_m = \frac{2}{3} \frac{0.001331 - 0.000166375}{0.0121 - 0.003025}$$ $$$$