To design a coupling by determining several parameters. We'll start with part (a): "Torque due to power delivered".

The formula for angular velocity $\omega$ in radians per second from revolutions per minute $N$ is:

$$\omega = \frac{2\pi N}{60}$$

Substitute the given value for $N$:

$$\omega = \frac{2\pi \times 800}{60} \text{ rad/s}$$ $$\omega = \frac{1600\pi}{60} \text{ rad/s}$$ $$\omega = \frac{80\pi}{3} \text{ rad/s}$$ $$\omega \approx 83.7758 \text{ rad/s}$$

The formula relating power, torque, and angular velocity is:

$$P = T\omega$$

Rearrange to solve for $T$:

$$T = \frac{P}{\omega}$$

Substitute the values from Step 1 and Step 2:

$$T = \frac{500 \times 10^3 \text{ W}}{83.7758 \text{ rad/s}}$$ $$T \approx 5968.31 \text{ N}\cdot\text{m}$$

The torque due to power delivered is approximately $5968.31 \text{ N}\cdot\text{m}$.