Problem 13 - Physics

Question

A person who weighs \(625 \mathrm{~N}\) is riding a 98-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 for each tire is approximately \(4.76 \times 10^{-4} \text{ m}^2\).

Step 1: Calculate Total Weight

First, find the total weight supported by the two tires. This includes both the weight of the person and the weight of the bike. The total weight is \(625 \text{ N} + 98 \text{ N} = 723 \text{ N}\).

Step 2: Determine Weight Supported by Each Tire

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

Step 3: Understand the Pressure Formula

Recall that pressure \(P\) is defined as force \(F\) divided by area \(A\), so \(P = \frac{F}{A}\). We can rearrange this formula to solve for area: \(A = \frac{F}{P}\).

Step 4: Calculate Area of Contact for One Tire

Substitute the force and pressure into the area formula: \(A = \frac{361.5 \text{ N}}{7.60 \times 10^{5} \text{ Pa}}\). Calculate the area to find \(A \approx 4.76 \times 10^{-4} \text{ m}^2\).

Key Concepts

Gauge Pressure

Gauge pressure refers to the pressure of a system or object in relation to atmospheric pressure. It's an important concept because it helps us understand the pressure exerted from the inside of a tire compared to the ambient atmospheric pressure outside.
Understanding gauge pressure starts with knowing that it is found by subtracting the atmospheric pressure from the absolute pressure of a system. Generally, the formula is expressed as \( P_{ ext{gauge}} = P_{ ext{absolute}} - P_{ ext{atmospheric}} \). In tire pressure scenarios, gauge pressure does not account for atmospheric pressure, since it's already present in our daily measurements.

• When your bike tires read a specific pressure like 760,000 Pa (Pascal), it actually refers to gauge pressure. This value doesn't include the atmospheric pressure, which is approximately 101,325 Pa at sea level.
• The concept of gauge pressure allows us to focus solely on the pressure that impacts the bike's tires exerting against the ground, ensuring the tires remain inflated to provide a smooth ride and proper load distribution.

Area of Contact

The area of contact is the specific part of the tire that touches the ground, and its measurement plays a crucial role in understanding how pressure and weight are managed.
In this context, the area of contact is important because it informs us how the weight of the rider and bike is distributed on the tire. Calculating the area of contact helps us make sure the tire can safely support the weight that is applied to it.

• The formula to find the area of contact is given by \( A = \frac{F}{P} \), where \( F \) is the force (or weight) and \( P \) is the pressure (gauge pressure of the tire).
• For the bike and rider example, we know the weight supported by each tire is 361.5 N and the gauge pressure for each tire is \( 7.60 \times 10^5 \) Pa.
• By substituting these values into the formula, you can find the area of contact for one tire. This gives us valuable insight into how the pressure and force interact physically at the point where the tire and ground meet.

Load Distribution

Load distribution relates to how weight or force is spread across a surface or various points. In our example, understanding load distribution is key to ensuring the weight of the rider and bike is properly carried by the tires.
The total weight of the rider and the bike is 723 N, which must be supported by both tires equally. This is why each tire bears half of the total weight, ensuring that the load is distributed evenly.

• This even load distribution is crucial for maintaining balance and stability while riding. Uneven distribution could lead to excessive wear on one tire or even unsafe riding conditions.
• By calculating each tire's load, using [total weight/2], we ensure that the pressure exerted through the tire onto the ground matches the optimal load distribution capacity of the tire.
• It's important to maintain proper tire inflation, which helps in achieving ideal load distribution, providing both safety and efficiency in movement.

Remember, maintaining equal weight distribution ensures a smoother ride and prolongs tire life by evenly distributing the wear and tear.