Problem 55 - College Physics

Question

The odometer (mileage gauge) of a car tells you the number of miles you have driven, but it doesn't count the miles directly. Instead, it counts the number of revolutions of your car's wheels and converts this quantity to mileage, assuming a standard-size tire and that your tires do not slip on the pavement. (a) A typical midsize car has tires 24 in. in diameter. How many revolutions of the wheels must the odometer count in order to show a mileage of 0.10 mile? (b) What will the odometer read when the tires have made 5000 revolutions? (c) Suppose you put oversize 28 -in.-diameter tires on your car. How many miles will you really have driven when your odometer reads 500 miles?

Step-by-Step Solution

Answer

(a) 84 revolutions; (b) 5.95 miles; (c) 584.63 miles.

Step 1: Find the Circumference of the Tire

The circumference of the tire is the distance the car travels in one revolution. For a tire of diameter 24 inches, the circumference \( C \) is calculated using the formula \( C = \pi \times d \), where \( d \) is the diameter.\[ C = \pi \times 24 \approx 75.4 \text{ inches} \]

Step 2: Convert Circumference to Miles

Since 1 mile equals 63,360 inches, convert the tire's circumference from inches to miles by dividing by 63,360.\[ C_{\text{miles}} = \frac{75.4}{63360} \approx 0.00119 \text{ miles/revolution} \]

Step 3: Calculate Revolutions for 0.10 Miles

To find the number of revolutions needed to cover 0.1 mile, divide 0.1 by the mileage per revolution.\[ \text{Revolutions} = \frac{0.1}{0.00119} \approx 84 \text{ revolutions} \]

Step 4: Calculate Mileage for 5000 Revolutions

To find the distance covered in miles for 5000 revolutions, multiply the number of revolutions by the mileage per revolution.\[ \text{Distance} = 5000 \times 0.00119 \approx 5.95 \text{ miles} \]

Step 5: Determine Actual Mileage with 28-inch Tires

For 28-inch diameter tires, first find the new circumference.\[ C' = \pi \times 28 \approx 87.96 \text{ inches} \]Convert to miles:\[ C'_{\text{miles}} = \frac{87.96}{63360} \approx 0.00139 \text{ miles/revolution} \]Then calculate the actual distance traveled when the odometer reads 500 miles by dividing 500 by the mileage per revolution of 24-inch tires and multiplying by that of the 28-inch tires.\[ \text{Revolutions (for 500 odometer miles)} = \frac{500}{0.00119} \approx 420168 \]Calculate the actual mileage with the new tires.\[ \text{Actual miles} = 420168 \times 0.00139 \approx 584.63 \text{ miles} \]

Key Concepts

Kinematics

Kinematics is a fundamental branch of physics that focuses on the motion of objects without considering the forces that cause these movements. It often uses concepts like displacement, velocity, and acceleration to understand motion.
In the odometer problem, we observe linear motion as the car travels along the road, converting circular wheel rotations into a linear distance. Each revolution of the car's tire translates into linear motion that the odometer measures in terms of miles traveled. To understand this, we must calculate how far the car moves for each complete tire rotation, which involves calculating the tire's circumference—the core of kinematics calculations in this context.

Tire Circumference Calculation

Calculating the circumference of a tire is crucial in determining how far a vehicle travels per wheel revolution. The formula to find the circumference is derived from the properties of a circle:

• The circumference (\( C \)) is calculated using the formula: \( C = \pi \times d \), where \(d\) is the diameter of the tire.
• For a 24-inch tire as given in the problem: \[ C = \pi \times 24 \approx 75.4 \text{inches} \]

This tells us how many inches the car will move forward per revolution of the tire. Understanding this fundamental calculation allows us to translate circular motion (revolutions) into linear motion (distance traveled).

Unit Conversion

Unit conversion plays a vital role in physics, especially when dealing with different measuring systems. To convert the tire's circumference from inches to miles, which is more practical for odometer readings:

• We know that 1 mile equals 63,360 inches.
• Using this, we convert the circumference:\[ C_{\text{miles}} = \frac{75.4}{63360} \approx 0.00119 \text{miles/revolution} \]

This conversion factor allows for determining how many miles are covered for each tire revolution and is pivotal for making accurate odometer readings.

Revolutions and Distance

Connecting revolutions of a tire to the actual distance driven requires using the circumference in miles per revolution.
This conversion enables two powerful calculations:

• To find the revolutions required for a certain distance (e.g., 0.10 miles): \[ \text{Revolutions} = \frac{0.1}{0.00119} \approx 84 \text{revolutions} \]
• To determine the distance for a given number of revolutions (e.g., 5000 revolutions): \[ \text{Distance} = 5000 \times 0.00119 \approx 5.95 \text{miles} \]

Utilizing these calculations ensures drivers understand how their vehicle's tires are influencing their odometer's indications.

Applied Physics

Applied physics takes theoretical concepts and uses them practically, like how changing tire size affects driven distance compared to odometer readings.

• For larger tires, like 28-inch ones, we first determine the new circumference: \[ C' = \pi \times 28 \approx 87.96 \text{inches} \]
• Convert this to miles: \[ C'_{\text{miles}} = \frac{87.96}{63360} \approx 0.00139 \text{miles/revolution} \]
• Calculate the actual miles driven when the odometer shows 500 miles, adjusting for the larger tire's circumference: \[ \text{Revolutions (for 500 odometer miles)} = \frac{500}{0.00119} \approx 420168 \]
• Then, actual miles for new tires: \[ \text{Actual miles} = 420168 \times 0.00139 \approx 584.63 \text{miles} \]

This clearly shows how real-life adjustments, like changing tire sizes, impact vehicle readings and measurements, demonstrating physics' practical applications.