Problem 57 - College Physics

Question

Odometer. 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 inches 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 \(5,000\) revolutions? (c) Suppose you put oversize 28 -inch-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) 210 revolutions; (b) 60.68 miles; (c) 583.33 miles

Step 1: Understand the Problem

First, recognize that the problem revolves around calculating the number of wheel revolutions required for a specific mileage and finding the actual mileage based on the number of wheel revolutions. We need to understand the relationship between the tire's diameter, its circumference, and how it relates to mileage.

Step 2: Calculate Tire Circumference (Part a)

The circumference of a tire is given by the formula \( C = \pi \times \text{diameter} \). For a 24-inch diameter tire, the circumference is \( C = \pi \times 24 \) inches.

Step 3: Convert Circumference to Miles

Since 1 mile equals 63,360 inches, we convert the tire's circumference from inches to miles: \( C = \frac{\pi \times 24}{63,360} \) miles per revolution.

Step 4: Determine Revolutions for 0.10 Mile (Part a)

To find the number of revolutions for 0.10 mile, use the formula: \( \text{Revolutions} = \frac{0.10}{C} \). Substitute \( C \) with the result from Step 3 to calculate the revolutions required.

Step 5: Read Odometer for 5,000 Revolutions (Part b)

With the circumference calculated in Step 3, calculate the distance covered in 5,000 revolutions: \( \text{Distance} = 5,000 \times C \). This gives the miles registered by the odometer.

Step 6: Recalculate with 28-inch Tires for Odometer 500 Miles (Part c)

For 28-inch diameter tires, recalculate the circumference \( C = \pi \times 28 \) inches, convert to miles, then use the distance formula: \( \text{Actual Miles} = 500 \times C \), where \( C = \frac{\pi \times 28}{63,360} \) miles per revolution.

Key Concepts

Tire Diameter

The diameter of a tire is the straight-line measurement from one side of the tire to the other, passing through the center. It is a crucial measurement because it directly affects how many revolutions a tire will make for a given distance.
In this exercise, the tire diameter is central to determining how far a car travels. For example, we deal with 24-inch and 28-inch tire diameters.


Having accurate tire diameter information is essential for understanding how an odometer calculates mileage and ensuring that the vehicle's speed and distance metrics are correct.

Wheel Revolutions

Wheel revolutions refer to the number of times a wheel spins completely around its axis during travel. This is an important concept because the odometer counts these revolutions to calculate the vehicle's traveled distance, assuming no slippage occurs between the tires and the road.

Each turn of the wheel moves the car forward by a distance equal to the tire's circumference. More revolutions mean the car has traveled a greater distance.
By knowing how many revolutions equate to specific mileage, one can determine how accurate the odometer's readings are under various circumstances, such as changes in tire size.

Mileage Calculation

Mileage calculation is how we determine the distance a vehicle has traveled. The odometer does this by using the number of wheel revolutions and converting this count into miles.

Here's the process for accurate mileage calculation:

• Calculate the tire's circumference.
• Determine how many miles each revolution covers (convert circumference from inches to miles).
• Multiply the number of revolutions by the distance covered per revolution to get total mileage.

This conversion is based on a standard tire, but changes in tire size can affect the accuracy unless adjustments are made.

Tire Circumference

The tire circumference is calculated using the formula for the circumference of a circle:
Circumference (C) = \(\pi \times \) Diameter.

For example, if the tire diameter is 24 inches, then the circumference is \(C = \pi \times 24\) inches.

This is the distance that the vehicle travels in one wheel revolution. Understanding tire circumference is key to converting revolutions into mileage.


Different diameters will yield different circumferences, which directly impacts the mileage computation.

Unit Conversion

Unit conversion is a necessary step when working with different measurement units, such as inches and miles.

In tire calculations, the circumference is initially in inches, but miles are the standard unit for odometer readings.

Therefore, we convert the circumference from inches to miles by using:
Circumference in miles = \(\frac{\pi \times \text{Diameter}}{63,360}\).

Knowing how to perform these conversions accurately ensures that odometer readings correctly reflect the distance the vehicle has traveled.
Understanding unit conversion is essential for resolving discrepancies which can arise when different tire sizes are used.