Let's solve this step-by-step.

Part (a): Angular Speed in rad/min

1. The blade spins at $1000$ revolutions per minute (rpm).
2. Since each revolution is $2\pi$ radians, the angular speed in radians per minute is calculated by: $\text{Angular Speed} = 1000 \times 2\pi = 2000\pi \, \text{rad/min}$

Answer for (a): The angular speed is $2000\pi$ rad/min.

Part (b): Linear Speed in ft/s

1. To find the linear speed $v$ of the sawteeth, we can use the formula: $v = r \cdot \omega$ where $r$ is the radius and $\omega$ is the angular speed in rad/min.

2. First, convert the radius from inches to feet: $r = 6 \, \text{in} = \frac{6}{12} \, \text{ft} = 0.5 \, \text{ft}$

3. Using $\omega = 2000\pi$ rad/min and $r = 0.5$ ft, the linear speed in feet per minute is: $v = 0.5 \times 2000\pi = 1000\pi \, \text{ft/min}$

4. Finally, to convert feet per minute to feet per second, divide by 60: $v = \frac{1000\pi}{60} \approx 52.36 \, \text{ft/s}$

Answer for (b): The linear speed is approximately $52.36 \, \text{ft/s}$.

Would you like further details on these calculations or have any questions?

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Here are five related questions to deepen understanding:

1. What would the angular speed be if the blade spun at 1200 rpm?
2. How would the linear speed change if the blade radius increased to 8 inches?
3. What is the relationship between linear and angular speed?
4. How would the answers differ if the blade radius were given in centimeters instead of inches?
5. Can angular speed in rad/min be converted to degrees per second?

Tip: Always double-check unit conversions, especially when switching between systems (e.g., inches to feet).