Problem 80 - Algebra and Trigonometry

Question

Radial Saw A radial saw has a blade with a 6 -in. radius. Suppose that the blade spins at 1000 \(\mathrm{rpm}\) . (a) Find the angular speed of the blade in rad/min. (b) Find the linear speed of the sawteeth in \(\mathrm{ft} / \mathrm{s}\) .

Step-by-Step Solution

Answer

(a) 2000\(\pi\) rad/min, (b) \(\approx 52.36\) ft/s

Step 1: Define Angular Speed

The angular speed \( \omega \) in radians per minute can be found using the formula \( \omega = 2 \pi \times \text{RPM} \). Since the blade spins at 1000 revolutions per minute (RPM), the angular speed \( \omega = 2 \pi \times 1000 = 2000\pi \) rad/min.

Step 2: Define Linear Speed Formula

The linear speed \( v \) is related to angular speed by the formula \( v = \omega \times r \), where \( r \) is the radius of the circle traced by the blade's edge.

Step 3: Calculate Linear Speed

Given the radius \( r = 6 \) inches and the angular speed from Step 1, substitute into the linear speed formula: \( v = 2000\pi \times 6 \). First, convert the radius into feet (since 1 foot = 12 inches), \( r = 0.5 \) feet. Therefore, \( v = 2000\pi \times 0.5 = 1000\pi \) ft/min.

Step 4: Convert Linear Speed into ft/s

Since the linear speed was calculated in feet per minute, convert it to feet per second by dividing by 60 (since 1 minute = 60 seconds). Thus, \( v = \frac{1000\pi}{60} \). Simplifying gives \( v \approx 52.36 \) ft/s.

Key Concepts

Angular Speed Calculation

Angular speed is a measure of how fast an object rotates or revolves relative to another point, usually the center of a circle. It is typically expressed in radians per minute (rad/min) or radians per second (rad/s), depending on the context.
For the radial saw with a blade radius of 6 inches, the angular speed can be calculated using the formula:

• degree of rotation: |
  \( \omega = 2\pi \times \text{RPM} \)

In our exercise, the blade spins at 1000 revolutions per minute (RPM). By substituting RPM into the formula, we find:

• \( \omega = 2\pi \times 1000 = 2000\pi \)

The result expresses the angular speed of the saw blade in radians per minute, indicating how quickly it spins around its axis. Remember, one complete revolution of the blade is equal to \( 2\pi \) radians.

Linear Speed Calculation

Linear speed refers to the distance an object travels per unit of time, moving in a straight line. In the context of circular motion, like that of a saw blade, linear speed can be found using the angular speed and the radius of the circular path.
To find the linear speed of the sawteeth, we use the relationship between angular speed \( \omega \) and linear speed \( v \):

• \( v = \omega \times r \)

Here, \( r \) is the radius of the blade's circular path. Initially, the radius is given as 6 inches, but for the purpose of finding linear speed in feet per second, we convert it to feet (\( r = 0.5 \) feet). By substituting the angular speed \( 2000\pi \) rad/min and the radius \( 0.5 \) ft into the formula:

• \( v = 2000\pi \times 0.5 = 1000\pi \) feet per minute

This gives the linear speed in feet per minute. It is a measure of how fast the sawteeth move along their circular path.

Unit Conversion in Physics

Unit conversion plays a crucial role in solving physics problems, especially when the given units don't match the desired unit of the answer. Here, we needed to convert linear speed from feet per minute to feet per second.
To convert from feet per minute (ft/min) to feet per second (ft/s), follow this simple step:

• Divide the speed in ft/min by 60, because there are 60 seconds in one minute.

For our linear speed of \( 1000\pi \) ft/min, the conversion is:

• \( v = \frac{1000\pi}{60} \approx 52.36 \) ft/s

This results in the linear speed in feet per second, making it more convenient for applications requiring a smaller time measurement. Such conversions are vital to ensure consistent units for understanding and solving physics problems accurately.