Problem 79 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Angular speed: 2000π rad/min. Linear speed: ≈ 52.36 ft/s.

Step 1: Understanding RPM

RPM stands for "Revolutions per minute." When the saw blade spins at 1000 RPM, it means the blade makes 1000 full rotations each minute.

Step 2: Convert RPM to rad/min

One full revolution corresponds to an angular displacement of \(2\pi\) radians. Therefore, to find the angular speed in rad/min, multiply the RPM by \(2\pi\):\[1000 \text{ RPM} \times 2\pi \text{ rad/rev} = 2000\pi \text{ rad/min}\].

Step 3: Calculate Linear Speed Formula

The linear speed \(v\) of any point on the edge of a rotating object can be expressed as the product of its radius \(r\) and its angular speed \(\omega\):\[v = r \times \omega\].

Step 4: Convert Radius to Feet

Since the radius of the blade is 6 inches, convert it to feet:\[6 \text{ inches} = \frac{6}{12} \text{ feet} = 0.5 \text{ feet}\].

Step 5: Calculate Linear Speed in ft/min

Use the formula from Step 3, substituting the radius and angular speed:\[v = 0.5 \text{ feet} \times 2000\pi \text{ rad/min} = 1000\pi \text{ ft/min}\].

Step 6: Convert Linear Speed to ft/s

Convert the linear speed from ft/min to ft/s by dividing by 60, as there are 60 seconds in a minute:\[v = \frac{1000\pi \text{ ft/min}}{60} = \frac{1000\pi}{60} \text{ ft/s} \approx 52.36 \text{ ft/s}\].

Key Concepts

Angular Speed

Angular speed describes how fast something rotates. It's often measured in radians per minute (rad/min) or radians per second (rad/s). A radian is a way to measure angles based on the circle's radius. Imagine the saw blade making circular rotations. Each full turn it makes is 360 degrees, which is the same as \(2\pi\) radians. So, when we talk about angular speed, we want to know how many radians the blade turns each minute.
To find the angular speed of the blade given its rotation rate of 1000 revolutions per minute (RPM), we multiply the revolutions by \(2\pi\). This tells us that 1000 revolutions are equivalent to \(2000\pi\) radians per minute. This helps us understand the spinning rate in terms of radians.

Linear Speed

Linear speed is the distance traveled by a point on the edge of a rotating object per unit time. In simpler terms, if you were to follow a point on the edge of the blade as it spins, linear speed would tell you how fast it’s moving in a straight line.
The formula to calculate linear speed (\( v \)) is:

• \( v = r \times \omega \)

Here, \( r \) is the radius of the circle (or the blade), and \( \omega \) is the angular speed in radians per minute. For instance, with our saw blade's radius at 0.5 feet and the angular speed as \(2000\pi\) rad/min, the linear speed would be \(1000\pi\) feet per minute before converting to other units like feet per second. Linear speed provides clarity on how fast the saw teeth move across materials they cut.

Unit Conversion

Unit conversion is a crucial aspect when dealing with measurements and speeds, especially when different units are involved. Understanding this helps translate between various measurements for better comprehension and real-world application.
In our example, we initially have the radius of the blade in inches. Since we often need to work in feet for linear speed, converting 6 inches to feet is necessary, resulting in 0.5 feet.

• 6 inches = \( \frac{6}{12} \) feet = 0.5 feet

Additionally, converting linear speeds from feet per minute to feet per second helps align with standard speed measurements used in practice. To do this conversion, divide the feet per minute value by 60 since there are 60 seconds in a minute, like converting \(1000\pi\) ft/min to approximately 52.36 ft/s. Unit conversion ensures consistency and understanding across different systems of measurements.

Revolutions per Minute (RPM)

Revolutions per minute, abbreviated as RPM, is a unit measuring the frequency of rotation. More simply, it indicates how many full spins an object, like a saw blade, completes in a single minute.
When someone mentions a saw blade spinning at 1000 RPM, it means every minute, the blade makes 1000 complete circles. This unit helps provide a standard way to communicate speed in contexts dealing with engines, turbines, or other spinning devices.
RPM is directly tied to angular speed since each revolution translates into an angular movement quantified in radians. This connection between RPM and angular speed can help bridge understanding between purely rotational speeds and speeds felt in linear or straight-line terms. RPM is essential for measuring and communicating performance in machinery and engineering.