Problem 84 - Precalculus Mathematics for Calculus

Question

A truck with 48-in.-diameter wheels is traveling at \(50 \mathrm{mi} / \mathrm{h}\). (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) 350.1 rad/min; (b) 55.7 rev/min

Step 1: Convert Speed to Inches per Minute

First, convert the truck's speed from miles per hour (mi/h) to inches per minute (in/min). Since the truck is traveling at 50 mi/h, remember that:- 1 mile = 5280 feet- 1 foot = 12 inchesTherefore, 1 mile = 5280 x 12 inches. Now, convert miles per hour to inches per minute:\[50 \text{ mi/h} = 50 \times 5280 \times 12 \text{ in/h} = 3168000 \text{ inches per hour}\]Since there are 60 minutes in an hour:\[3168000 \div 60 = 52800 \text{ in/min}\]

Step 2: Find the Circumference of the Wheel

To find the angular speed, first calculate the circumference of the wheel, which is a circle. Use the formula for the circumference of a circle, which is:\[C = \pi \times d\]where \(d\) is the diameter. Here, the diameter is 48 inches, so:\[C = \pi \times 48 \approx 150.8 \text{ in}\]

Step 3: Calculate Angular Speed in Radians per Minute

Now, calculate the angular speed in radians per minute. Use the relationship between linear speed and angular speed:\[\text{Angular Speed} = \frac{\text{Linear Speed}}{\text{Circumference}}\]Substitute the values we’ve calculated:\[\text{Angular Speed} = \frac{52800 \text{ in/min}}{150.8 \text{ in}} \approx 350.1 \text{ rad/min}\]

Step 4: Calculate Revolutions per Minute

Finally, find how many revolutions per minute the wheels make. Remember, one full revolution is \(2\pi\) radians. Therefore, the number of revolutions is:\[\text{Revolutions per minute} = \frac{\text{Angular Speed in rad/min}}{2\pi}\]Substitute the angular speed:\[\text{Revolutions per minute} = \frac{350.1}{2\pi} \approx 55.7 \text{ rev/min}\]

Key Concepts

Radians per Minute

When dealing with circular motion, understanding angular speed in terms of radians per minute is crucial. Angular speed describes how fast an angle is changing in a rotating system. In the context of wheels, it tells us how fast the wheel is spinning in relation to a fixed central point. This is typically represented in radians per unit of time.

To convert linear speed (such as mph) to angular speed in radians per minute, you first need to consider the distance traveled per unit time. Here, this was converted from miles per hour to inches per minute for easier calculation since wheel diameters are often given in smaller units like inches.

Once you have that linear speed in inches per minute, you need to know the circumference of the wheel. You then divide the linear speed by the circumference of the wheel. This calculation gives you the number of radians the wheel turns through per minute, as shown:

• Calculate total distance per minute (inches per minute),
• Identify the circumference of the wheel (inches),
• Use the formula: Angular Speed = Linear Speed / Circumference.

This process results in a measure of radians per minute, allowing us to understand how fast the wheel is rotating at its core.

Revolutions per Minute

Revolutions per minute (RPM) gives a more familiar measurement of how frequently an object completes a full circle or cycle in one minute. It's commonly used in describing how fast a wheel or any rotational body moves.

To find RPM from radians per minute, it's vital to remember that one complete revolution is equivalent to an angle of \(2\pi\) radians. Therefore, you take the angular speed in radians per minute and divide it by \(2\pi\). This calculation converts the angular speed into revolutions:

• Start with Angular Speed in radians per minute,
• Divide by \(2\pi\) (since \(2\pi\) radians equals one full revolution),
• Result is the RPM, representing full wheels rotations per minute.

This makes it easier to visualize how many times the wheel spins around in a minute, linking the idea of angular movement back to a complete, understandable cycle.

Circumference of a Circle

The circumference of a circle is the linear distance around its edge. It’s a crucial component in calculating both angular speed and revolutions. For any wheel, including the truck's in the problem, knowing the circumference allows you to relate linear speed to rotation.

The formula for calculating a circle's circumference is arranged simply as:

• \(C = \pi \times d\)

where \(d\) is the diameter of the circle. In this specific exercise, the wheel’s diameter was 48 inches, leading to a calculation of:

• \(C = \pi \times 48 \),
• This equates to approximately 150.8 inches.

Having the correct circumference is essential because it directly relates the linear travel distance of a point on the wheel's edge to how far the wheel turns around its axis. This result is then used to find both the angular speed (in radians per minute) and revolutions per minute. Understanding the circumference connects the physical size of the wheel with its rotational motion.