Problem 13 - Physics

Question

A person who weighs \(625 \mathrm{~N}\) is riding a \(98-\mathrm{N}\) mountain bike. Suppose the entire weight of the rider and bike is supported equally by the two tires. If the gauge pressure in each tire is \(7.60 \times 10^{5} \mathrm{~Pa}\), what is the area of contact between each tire and the ground?

Step-by-Step Solution

Answer

The area of contact between each tire and the ground is approximately \(4.755 \times 10^{-4} \mathrm{~m}^2\).

Step 1: Calculate Total Weight

First, add the weight of the person and the mountain bike to find the total weight supported by the tires. The total weight is \(625 \mathrm{~N} + 98 \mathrm{~N} = 723 \mathrm{~N}\).

Step 2: Weight Supported by Each Tire

Since the weight is equally supported by the two tires, divide the total weight by 2 to find the weight supported by each tire. \(\frac{723 \mathrm{~N}}{2} = 361.5 \mathrm{~N}\).

Step 3: Use the Pressure Formula

The pressure formula is given by \(P = \frac{F}{A}\), where \(P\) is the pressure, \(F\) is the force (weight supported by each tire), and \(A\) is the contact area. We need to solve for \(A\).

Step 4: Solve for Area

Rearrange the pressure formula to solve for \(A\): \[ A = \frac{F}{P} \]Substitute the known values: \[ A = \frac{361.5 \mathrm{~N}}{7.60 \times 10^{5} \mathrm{~Pa}} \]

Step 5: Calculate Contact Area

Perform the calculation to find the contact area: \[ A = \frac{361.5}{7.60 \times 10^{5}} \approx 4.755 \times 10^{-4} \mathrm{~m}^2 \].

Key Concepts

Contact Area

When dealing with tires, the term "contact area" refers to the portion of the tire that is in direct contact with the ground. This area is crucial for several reasons. It impacts how the tire grips the road, affects the ride smoothness, and influences the wear rate of the tire.

In our problem, we found the contact area from the given pressure and weight. By using the pressure formula, we determined how much "flat" surface of the tire touches the road under the given load. A smaller contact area implies that the tire is well-rounded and more pressurized, while a larger area might suggest a flatter, under-inflated tire.

Understanding contact area helps with tire maintenance. For example, checking if the tires are adequately inflated is essential for safety and efficiency. Both underinflated and overinflated tires change the contact area, which can lead to reduced stability and increased tire wear.

Gauge Pressure

Gauge pressure is what we often refer to when talking about tire pressure, essentially measuring the pressure in the tire minus atmospheric pressure. This is different from absolute pressure, which includes atmospheric pressure. In practical terms, when you use a tire pressure gauge, you're measuring gauge pressure.

This pressure is pivotal in our exercise, as it helps us calculate the contact area of the tires. The gauge pressure in each tire is given as \(7.60 \times 10^5 \text{ Pa}\). Knowing this, we can determine how much force per unit area the tire can exert on the ground.

Proper gauge pressure ensures that tires have the correct contact area, enhancing vehicle performance. If the gauge pressure is too high, the contact area is reduced, which can lead to less grip. Conversely, if it's too low, the increased contact area might mean better grip but also more frictional losses and unusual wear patterns.

Force and Area Relationship

The relationship between force and area is central to understanding the concepts of pressure and contact areas. Pressure is defined as the force applied per unit area, which can be written mathematically as: \[ P = \frac{F}{A} \]Here, \(P\) is the pressure, \(F\) is the force applied, and \(A\) is the area over which the force acts.

In our exercise, the problem uses this relationship to determine the contact area of a bike tire. The equation is rearranged to solve for the area \(A\), giving:\[ A = \frac{F}{P} \]Given that each tire supports a force of \(361.5 \text{ N}\), and knowing the gauge pressure, we calculated the contact area needed to support this force.

Understanding this relationship is vital for resolving many real-world scenarios, such as designing tires to distribute forces effectively, ensuring safety, and optimizing grip. This knowledge assists in making informed decisions about the consumption of products and their maintenance.