A miter saw has a blade with a 7-in. radius. Suppose that the blade spins at 5,000 rpm.
(a)
Find the angular speed of the blade in rad/min.

(b)
Find the linear speed of the sawteeth in ft/s.

Let's break down the problem into two parts:

### Part (a): Angular Speed of the Blade in rad/min
The blade spins at 5,000 revolutions per minute (rpm). We want to convert this angular speed to radians per minute (rad/min).

1. **Conversion of Revolutions to Radians**:
   - One complete revolution is \(2\pi\) radians.
   - So, 5,000 revolutions per minute is equivalent to:
   \[
   \text{Angular Speed} = 5000 \times 2\pi \, \text{rad/min}
   \]
   \[
   \text{Angular Speed} = 10000\pi \, \text{rad/min}
   \]

So, the angular speed of the blade is \(10,000\pi\) rad/min.

### Part (b): Linear Speed of the Sawteeth in ft/s
Now, we want to find the linear speed of the sawteeth in feet per second (ft/s).

2. **Finding the Circumference**:
   - The radius of the blade is given as 7 inches.
   - The linear speed \(v\) at the edge of the blade (which is the speed of the sawteeth) is related to the angular speed \(\omega\) by the formula:
   \[
   v = r \omega
   \]
   where \(r\) is the radius in feet, and \(\omega\) is the angular speed in radians per second.

3. **Convert Radius to Feet**:
   \[
   r = \frac{7 \, \text{inches}}{12 \, \text{inches/foot}} = \frac{7}{12} \, \text{feet}
   \]

4. **Convert Angular Speed to rad/s**:
   - The angular speed in rad/min was found to be \(10,000\pi\) rad/min.
   - Convert this to rad/s by dividing by 60:
   \[
   \omega = \frac{10000\pi \, \text{rad/min}}{60 \, \text{sec/min}} = \frac{1000\pi}{6} \, \text{rad/s} = \frac{500\pi}{3} \, \text{rad/s}
   \]

5. **Calculate Linear Speed**:
   \[
   v = r \omega = \left(\frac{7}{12}\right) \times \left(\frac{500\pi}{3}\right) \, \text{ft/s}
   \]
   \[
   v = \frac{7 \times 500\pi}{36} \, \text{ft/s}
   \]
   \[
   v = \frac{3500\pi}{36} \, \text{ft/s}
   \]
   \[
   v \approx 305.5 \, \text{ft/s}
   \]

So, the linear speed of the sawteeth is approximately **305.5 ft/s**.

### Summary:
- (a) The angular speed of the blade is \(10,000\pi\) rad/min.
- (b) The linear speed of the sawteeth is approximately **305.5 ft/s**.

Do you want further details or have any questions?

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Here are five related questions to consider:
1. How would the linear speed change if the blade radius was doubled?
2. What would the linear speed be if the blade spun at 6,000 rpm instead of 5,000 rpm?
3. How does the linear speed relate to the cutting power of the saw?
4. How can you convert the linear speed into miles per hour (mph)?
5. What is the difference between angular speed and linear speed in practical terms?

**Tip:** When dealing with rotational motion, always remember that the angular speed (rad/s) can be converted to linear speed (m/s) using the radius of the circular path.

Learn how to calculate the angular speed and linear speed of a miter saw blade with a 7-inch radius spinning at 5,000 rpm. Understand the conversion from revolutions per minute (rpm) to radians per minute (rad/min) for angular speed, and determine the linear speed of the sawteeth in feet per second (ft/s).

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