Problem 80 - Precalculus Mathematics for Calculus

Question

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 \(\mathrm{ft} / \mathrm{s}\).

Step-by-Step Solution

Answer

(a) \(2000\pi\) rad/min; (b) \(\frac{50\pi}{3}\) ft/s or approximately 52.36 ft/s.

Step 1: Understand Angular Speed

Angular speed is the angle through which a point or line has rotated in a given time period. It is measured in radians per minute (or similar units). Since 1 rotation is equal to \(2\pi\) radians and the blade spins at 1000 revolutions per minute, the angular speed in rad/min can be found by multiplying the number of revolutions per minute by \(2\pi\).

Step 2: Compute Angular Speed

To find the angular speed in rad/min, multiply the revolutions per minute by \(2\pi\):\[\text{Angular Speed} = 1000 \times 2\pi = 2000\pi \text{ rad/min}\]

Step 3: Understand Linear Speed

Linear speed is the distance traveled per unit of time along a path. For a point on a rotating object, like the sawteeth, it can be found by multiplying the angular speed by the radius of the circle (blade). Convert the radius to feet and the time to seconds to match the desired units (ft/s).

Step 4: Convert Radius to Feet

The radius of the blade is given in inches (6 inches). First, convert this to feet, knowing there are 12 inches in a foot:\[\text{Radius in feet} = \frac{6}{12} = 0.5 \text{ feet}\]

Step 5: Convert Angular Speed to Correct Units

The angular speed is given in rad/min. First, we'll need the speed in rad/s to find the linear speed in ft/s. There are 60 seconds in a minute:\[\text{Angular Speed in rad/s} = \frac{2000\pi}{60} = \frac{100\pi}{3} \text{ rad/s}\]

Step 6: Compute Linear Speed

Multiply the angular speed in rad/s by the radius in feet:\[\text{Linear Speed} = \left(\frac{100\pi}{3}\right) \times 0.5 = \frac{50\pi}{3} \text{ ft/s}\]

Step 7: Finalize Answers

Now, finalize and summarize the answers:(a) The angular speed of the blade is \(2000\pi\) rad/min.(b) The linear speed of the sawteeth is approximately \(52.36\) ft/s (since \(\pi \approx 3.1416\)).

Key Concepts

linear speed

Linear speed refers to how fast an object travels along a path. It is an important concept in rotational motion. If you think about a rotating wheel, linear speed pertains to a point on the edge of the wheel.

• To calculate the linear speed of a rotating object, you multiply its angular speed by its radius.
• For instance, if a blade rotates, the linear speed of any point on the edge is the angular speed (in rad/s) times the radius (in feet, meters, etc.).

In the given exercise, first, the radius of the blade was converted from inches to feet. The radius in feet is 0.5 ft. Next, the linear speed was calculated using the angular speed in rad/s. This conversion is vital when wanting results in a specific unit of measurement, like ft/s.
The linear speed tells you how quickly a particular point, such as a saw tooth on the blade, is moving.

revolutions per minute (rpm)

Revolutions per minute (rpm) is a unit of rotational speed. It measures how many complete rotations an object makes in one minute. When we talk about rotating machinery or even parts like saw blades, rpm is a handy measurement.

• For example, if a blade spins at 1000 rpm, it completes 1000 full circles every minute.
• Rpm is useful for understanding how fast something is spinning over time.

Why is this important for the problem at hand? In our exercise, the saw blade spins at 1000 rpm, and this is converted into angular speed by using the relationship that 1 revolution equals \(2\pi\) radians. Hence, the angular speed in rad/min is calculated by multiplying 1000 rpm with \(2\pi\). This conversion is vital to determine how fast the blade is moving in terms of radians per unit time, which is useful for overall speed calculations.

radian

A radian is a unit of angular measure. Unlike degrees, which divide a circle into 360 parts, radians use the radius of the circle to define angles. One full rotation around a circle is \(2\pi\) radians.

• Radians provide a natural way to express angles as they relate directly to the arc length of a circle.
• This relationship with the circle's geometry makes radians invaluable in mathematics and physics involving rotational motion.

In the context of our problem, converting rpms to rad/min involves multiplying the revolutions by \(2\pi\) because each revolution covers the full circle—\(2\pi\) radians. Addressing the angular speed in radians helps connect linear measurements on the rotating saw blade. Understanding radians simplifies calculating angular and linear speeds with consistent accuracy.
Radians ensure that calculations of motion on circular paths remain intuitive and straightforward.