The circular blade on a saw has a diameter of 7.5 inches and rotates at 2400 revolutions per minute find the linear speed in feet per second

To find the linear speed (or tangential speed) of a point on the edge of the circular blade, we can use the following formula:

\[
v = \omega \cdot r
\]

where:
- \( v \) is the linear speed,
- \( \omega \) is the angular velocity in radians per second,
- \( r \) is the radius of the blade.

### Step 1: Convert the diameter to radius
The diameter of the blade is 7.5 inches, so the radius is:

\[
r = \frac{7.5}{2} = 3.75 \text{ inches}.
\]

We will need to convert inches to feet since the final answer should be in feet per second:

\[
r = \frac{3.75}{12} = 0.3125 \text{ feet}.
\]

### Step 2: Convert revolutions per minute (RPM) to radians per second
The angular velocity in radians per second can be found by converting the RPM. One revolution is \(2\pi\) radians, and 1 minute is 60 seconds, so:

\[
\omega = 2400 \times \frac{2\pi}{60} = 2400 \times \frac{\pi}{30}.
\]

### Step 3: Calculate the linear speed
Now, we can calculate the linear speed:

\[
v = \omega \cdot r = \left(2400 \times \frac{\pi}{30}\right) \cdot 0.3125.
\]

Let's calculate the value of \( v \).The linear speed of the circular blade is approximately \( 78.54 \) feet per second.

Would you like further details or have any other questions? Here are some related questions that build on this topic:

1. How would the linear speed change if the diameter of the blade was 10 inches?
2. What is the relationship between angular speed and linear speed in circular motion?
3. How would a decrease in revolutions per minute affect the blade's linear speed?
4. Can we calculate the total distance traveled by a point on the blade after 5 seconds?
5. What would the linear speed be in miles per hour?

**Tip:** To convert between angular and linear speeds, always keep track of units (inches, feet, seconds, etc.) to ensure accuracy.

The circular blade on a saw has a diameter of 7.5 inches and rotates at 2400 revolutions per minute. Find the linear speed in feet per second.

Learn how to calculate the linear speed of a circular saw blade with a diameter of 7.5 inches rotating at 2400 RPM. This problem uses key geometric and circular motion concepts, including unit conversions, angular velocity, and linear speed, suitable for Grades 9-12.

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