Problem 82 - Algebra and Trigonometry

Question

Truck Wheels A truck with 48-in.-diameter wheels is traveling at 50 mi/h. (a) Find the angular speed of the wheels in rad/min. (b) How many revolutions per minute do the wheels make?

Step-by-Step Solution

Answer

(a) 2200 rad/min; (b) 350.37 rev/min.

Step 1: Convert Linear Speed to Inches per Minute

The truck is traveling at a speed of 50 miles per hour. First, convert this speed to inches per minute. 1 mile = 5280 feet, and 1 foot = 12 inches, so 1 mile = 5280 x 12 = 63360 inches.The truck's speed in inches per hour is:\[50 \times 63360 = 3168000 \text{ inches per hour}.\]Now, convert this to inches per minute by dividing by 60:\[\frac{3168000}{60} = 52800 \text{ inches per minute}.\]

Step 2: Calculate Angular Speed in Radians per Minute

The diameter of the wheels is 48 inches, so the radius \(r\) is half of that:\[ r = \frac{48}{2} = 24 \text{ inches}. \]The angular speed \(\omega\) in radians per minute is given by \(\omega = \frac{v}{r}\), where \(v\) is the linear speed.Substitute \(v = 52800\) inches per minute and \(r = 24\) inches to find \(\omega\):\[\omega = \frac{52800}{24} = 2200 \text{ radians per minute}.\]

Step 3: Convert Angular Speed to Revolutions per Minute

One complete revolution of the wheel corresponds to \(2\pi\) radians. To find the number of revolutions per minute, divide the angular speed by \(2\pi\):\[\text{Revolutions per minute} = \frac{2200}{2\pi} \approx 350.37 \text{ rev/min}.\]

Key Concepts

Linear Speed Conversion

Linear speed conversion involves translating a given speed into a more conveniently used unit. In the exercise, we start with 50 miles per hour and aim to express this speed in inches per minute.

First, understand the conversion factors: 1 mile equals 5280 feet, and 1 foot equals 12 inches. So, 1 mile can be converted into 63360 inches. This result is obtained by multiplying 5280 by 12.

To convert the truck's speed from miles per hour to inches per hour, multiply 50 by 63360, resulting in 3168000 inches per hour. To change this into inches per minute, divide by 60 because there are 60 minutes in an hour. Hence, the speed is 52800 inches per minute.

Converting speeds allows us to tackle problems involving rotations or angular movements effectively, by aligning linear measurements with rotational quantities.

Wheel Revolutions

Understanding wheel revolutions is crucial when dealing with rotation problems. Revolutions per minute (RPM) measure how many complete turns a wheel makes in one minute.

First, determine the angular speed in radians per minute, and then convert this to revolutions. As previously calculated, the angular speed is 2200 radians per minute.

Since each full wheel revolution equals to a complete circle which is defined mathematically as a rotation of \(2\pi\) radians, you divide the angular speed by \(2\pi\) to find out the number of revolutions.

Thus, the number of wheel revolutions per minute is \(\frac{2200}{2\pi} \approx 350.37\). This approach links linear motion with rotational motion, providing a full picture of the dynamics at play.

Radian Measurement

Radians are the standard unit of angular measure used in many areas, especially in mathematics and physics. A radian measures the angle created when a radius is wrapped along the circle's edge.

In practical terms, one complete revolution around a circle corresponds to \(2\pi\) radians. This fact aids in converting between angles in radians and rotations or revolutions, as seen in the exercise.

The angular speed in radians is beneficial as it allows us to convert linear speeds to rotational dynamics effortlessly. It is expressed as \(\omega = \frac{v}{r}\) where \(v\) is the linear speed, and \(r\) is the radius of the circle.

Understanding radians equips students with the tools to handle a wide range of problems involving circular motion, and it simplifies the calculations connecting circular and linear dynamics.

Diameter and Radius Calculation

Diameter and radius are fundamental measurements of a circle, crucial for determining the wheel's characteristics.

The diameter of the wheel is the total length across the circle through the center. In the problem, the diameter is given as 48 inches. For most rotational calculations, the radius is more useful. The radius, half the diameter, is the distance from the circle’s center to any point on its circumference.

Thus, the radius of the wheel is \(\frac{48}{2} = 24\) inches.

The radius is essential because it serves as a key variable in the formula for angular speed \(\omega = \frac{v}{r}\), where it provides the necessary connection between linear and rotational movements. Understanding the relationship between diameter and radius simplifies circular motion problems and is crucial for anyone delving into physics or engineering.