Problem 58 - Precalculus

Question

A truck with \(32\)-inch diameter wheels is traveling at \(60\) mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

Step-by-Step Solution

Answer

The angular speed is approximately 630.86 rad/min, and the wheels make about 100.42 revolutions per minute.

Step 1: Convert Wheel Diameter to Radius

The diameter of the truck wheels is given as \(32\) inches. To find the radius, divide the diameter by \(2\):\[ r = \frac{32}{2} = 16 \text{ inches} \]

Step 2: Convert Speed to Inches per Minute

The truck's speed is \(60\) miles per hour. First, convert miles to inches:\[ 1 \text{ mile} = 5280 \times 12 = 63360 \text{ inches} \]Thus, \(60\) mi/h is:\[ 60 \times 63360 = 3801600 \text{ inches per hour} \]Next, convert the speed to inches per minute:\[ \frac{3801600}{60} = 63360 \text{ inches per minute} \]

Step 3: Calculate Angular Speed in Rad/Min

The circumference of a wheel is the distance covered in one revolution, calculated as:\[ C = 2\pi r = 2\pi \times 16 = 32\pi \text{ inches} \] The angular speed \(\omega\) in radians per minute is the ratio of the linear speed to the circumference:\[ \omega = \frac{63360}{32\pi} \approx \frac{1980}{\pi} \approx 630.86 \text{ rad/min} \]

Step 4: Calculate Revolutions Per Minute (RPM)

Since one full revolution corresponds to \(2\pi\) radians, calculate the number of revolutions per minute:\[ \text{RPM} = \frac{\omega}{2\pi} = \frac{630.86}{2\pi} \approx 100.42 \text{ revolutions per minute} \]

Key Concepts

Radius Conversion

Understanding how to convert diameter to radius is crucial, as many calculations in mathematics and physics rely on the radius. When given the diameter, which is the total distance across a circle through its center, you can find the radius by simply dividing the diameter by 2. For instance, if a truck wheel has a diameter of 32 inches, the radius would be calculated as follows:\[ r = \frac{32}{2} = 16 \text{ inches} \]This basic conversion is essential for more advanced calculations, such as finding the wheel's circumference or determining angular speed. Remember, the radius is always half the diameter, making it a straightforward process.

Speed Conversion

Converting speed from one unit to another can seem tricky, but it's quite simple once you understand the basic steps. For instance, when a truck is moving at 60 miles per hour (mi/h) and you need to convert this speed to inches per minute, start by understanding the conversion factors:

• 1 mile equals 5280 feet.
• 1 foot equals 12 inches.
• 1 hour equals 60 minutes.

Using these, convert the truck's speed:1. Convert miles to inches: \[ 1 \text{mile} = 5280 \times 12 = 63360 \text{inches} \]2. Then calculate the speed in inches per hour: \[ 60 \times 63360 = 3801600 \text{ inches/hour} \]3. Finally, convert to inches per minute: \[ \frac{3801600}{60} = 63360 \text{ inches/minute} \]Breaking down the steps makes it easier to see how units change through the conversion.

Circumference Calculation

The circumference of a wheel, or circle, is the perimeter or distance all the way around it. Calculating the circumference is essential because it tells us how far the wheel travels in one complete rotation. The formula used is:\[ C = 2\pi r \]In this scenario, with a radius of 16 inches, the circumference is calculated as:\[ C = 2\pi \times 16 = 32\pi \text{ inches} \]This result indicates the wheel travels 32π inches in one full turn. Knowing the circumference is critical for finding the angular speed and revolutions per minute, as it relates the linear motion of the wheel on the road to its rotational motion.

Revolutions Per Minute

Revolutions per minute (RPM) measures how many times an object, like a wheel, spins in one minute. To calculate RPM, start with the angular speed in radians per minute, which can be derived from the linear speed and the wheel's circumference:\[ \omega = \frac{63360}{32\pi} \approx 630.86 \text{ rad/min} \]Since one full revolution equals \(2\pi\) radians, convert the angular speed to RPM:\[ \text{RPM} = \frac{630.86}{2\pi} \approx 100.42 \text{ revolutions per minute} \]This shows how often the wheel completely turns within one minute. RPM is a practical measurement often used to describe the speed of rotating objects like wheels, engines, or disks.