To solve this problem, let us calculate the linear speed of the point of a tooth on the saw blade in miles per hour.

Step 1: Understand the relationships

• The diameter of the blade is $8 \, \text{inches}$, so the radius is: $r = \frac{\text{diameter}}{2} = \frac{8}{2} = 4 \, \text{inches}.$

• The blade rotates at $5000 \, \text{rpm}$ (revolutions per minute).

• We need to calculate the linear speed at the outer edge (tooth point): $v = \omega \cdot r,$ where:

  • $\omega$ is the angular velocity in radians per second,
  • $r$ is the radius.

Step 2: Convert $\text{rpm}$ to radians per second

• Angular velocity in radians per second: $\omega = 5000 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{\text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}}.$ Calculate: $\omega = 5000 \times \frac{2\pi}{60} = 5000 \times \frac{\pi}{30}.$

Step 3: Calculate the linear speed in inches per second

• Linear speed in inches per second: $v = \omega \cdot r = \left( 5000 \times \frac{\pi}{30} \right) \cdot 4.$ Simplify: $v = 5000 \times \frac{\pi}{30} \cdot 4 = 5000 \times \frac{4\pi}{30}.$ $v = \frac{20000\pi}{30} \, \text{inches per second}.$

Step 4: Convert to miles per hour

• First, convert inches per second to miles per hour: $1 \, \text{mile} = 63360 \, \text{inches}, \quad 1 \, \text{hour} = 3600 \, \text{seconds}.$ Therefore: $\text{Speed in mph} = v \cdot \frac{1 \, \text{mile}}{63360 \, \text{inches}} \cdot 3600 \, \text{seconds per hour}.$ Substitute $v = \frac{20000\pi}{30}$: $\text{Speed in mph} = \frac{20000\pi}{30} \cdot \frac{1}{63360} \cdot 3600.$

Simplify: $\text{Speed in mph} = \frac{20000\pi \cdot 3600}{30 \cdot 63360}.$ $\text{Speed in mph} = \frac{72000000\pi}{1900800}.$ $\text{Speed in mph} \approx \frac{226194.67}{1900800}.$ $\text{Speed in mph} \approx 68 \, \text{mph}.$

Final Answer:

The linear speed at the point of a tooth on the blade is approximately 68 mph.

Would you like me to explain any step in more detail or clarify further?

Related Questions:

1. What is the formula for converting angular velocity to linear velocity?
2. How can you calculate angular velocity from revolutions per minute (rpm)?
3. How do you convert inches per second to miles per hour?
4. Why is the factor $2\pi$ used in angular velocity calculations?
5. Can the same calculation process be used for other circular tools, like wheels?

Tip:

Always check your units carefully when performing multi-step calculations involving conversions, especially for angular and linear velocities.