Problem 92 - Precalculus : Building Concepts and Connections

Question

A car with tires 24 inches in diameter travels at a specd of 60 mph. What is the angular speed of the tires? Express your answer in both degrees per second and radians per second.

Step-by-Step Solution

Answer

The angular speed of the tires is 1500 Degrees per Second and approximately 26.18 Radians per Second.

Step 1: Calculate the Velocity in Inches per Second

First, the speed needs to be converted from miles per hour to inches per second, because the diameter of the tire is given in inches. So, convert the speed from mph to inches per second using the conversion factors 5280 feet in 1 mile and 12 inches in 1 foot. The speed in inches per second is \(60 \times 5280 \times 12/3600\).

Step 2: Calculate the Car's Angular Speed in Degrees per Second

The tire makes one complete revolution (360 degrees) for a distance equals its circumference. The circumference of the tire is \(pi \times\) Diameter. So, the angular speed in degrees per second is (Speed in inches per second) / (tire's circumference) \times 360. This results in \((60 \times 5280 \times 12/3600) / (pi \times 24) \times 360\) degrees per second.

Step 3: Convert Degrees to Radians

We know that \(2 \pi\) radians equal 360 degrees. So, we can convert the angular speed from degrees per second to radians per second by multiplying the speed in degrees per second by \(2 \pi/360\). Hence, the angular speed in radians per second is \((60 \times 5280 \times 12/3600) / (pi \times 24) \times 360 \times 2 \pi/360\). This simplifies to \((60 \times 5280 \times 12 \times 2) / (24^2) \) radians per second.

Key Concepts

Converting Speeds

Understanding how to convert speeds from units like miles per hour (mph) to inches per second (in/s) is crucial for solving problems involving motion. First, it's important to identify the conversion factors that will be used: there are 5280 feet in a mile and 12 inches in one foot. In the exercise, we're converting 60 mph to in/s.

Here's the process in detail: Multiply the speed in mph by the number of feet in a mile (5280) to convert miles to feet, and then by the number of inches in a foot (12) to convert feet to inches. Since there are 3600 seconds in an hour (60 minutes/hour times 60 seconds/minute), we then divide by 3600 to make the final conversion from hours to seconds.

Circumference of a Circle

The circumference of a circle is the distance around it, and it's important for problems related to rotary motion, like finding a tire's angular speed. The formula for the circumference of a circle is \( C = \pi \times d \) where \(C\) is the circumference and \(d\) is the diameter of the circle. Given a tire diameter of 24 inches, the circumference would be \(\pi \times 24\) inches.

The exercise shows that the distance the car covers in one revolution of the tire is directly related to the circumference. Since we know the tire’s linear speed in inches per second (from the speed conversion), we can determine how many degrees the tire rotates in one second by dividing this by the tire's circumference and then multiplying by 360 degrees, which is the total number of degrees in a full rotation.

Degrees to Radians Conversion

When working with angular measures, it's useful to convert between degrees and radians. Here's the essential conversion factor to remember: \(360^\circ = 2\pi\) radians. So, to convert degrees to radians, we use the ratio of \(2\pi\) radians over 360 degrees.

Take the angular speed we found in degrees per second from the previous step. By multiplying by \(2\pi/360\), we're effectively changing the unit from degrees to radians, because each degree is converted into its radian equivalent. The conversion simplifies the expression, as the 360s cancel out, and we're left with the tire's angular speed in radians per second. This is essential for various physics calculations, particularly when you're working with angular velocities in equations that require the angular speed to be in radians per second.