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Chapter 5: Problem 34

A nail whose cross-sectional area is \(3 \mathrm{mm}^{2}\) is embedded in a tire in which the air pressure is 1.8 bar. How much force tends to push the nail out?

Short Answer

Expert verified
0.54 N

Step by step solution

01

Understand the given information

The problem provides a nail with a cross-sectional area of \(3 \, \mathrm{mm}^{2}\) embedded in a tire. The tire has an internal air pressure of \(1.8 \, \mathrm{bar}\).
02

Convert units if necessary

Convert the pressure from bar to the standard unit, Pascals (Pa). \(1 \, \mathrm{bar} = 100,000 \, \mathrm{Pa}\). Thus, \(1.8 \, \mathrm{bar} = 1.8 \, \times \, 100,000 \, \mathrm{Pa} = 180,000 \, \mathrm{Pa}\).
03

Convert area to square meters

Convert the cross-sectional area from \(\mathrm{mm}^{2}\) to \(\mathrm{m}^{2}\). Since \(1 \, \mathrm{mm} = 0.001 \, \mathrm{m}\): \[3 \, \mathrm{mm}^{2} = 3 \, \times \, (0.001 \, \mathrm{m})^{2} = 3 \, \times \, 10^{-6} \, \mathrm{m}^{2}\]
04

Use the pressure-area relationship

The force exerted by the pressure on the area is given by \( F = P \, \times \, A \).
05

Calculate the force

Use the values: \(P = 180,000 \, \mathrm{Pa}\) and \(A = 3 \, \times \, 10^{-6} \, \mathrm{m}^{2}\). \[F = 180,000 \, \mathrm{Pa} \, \times \, 3 \, \times \, 10^{-6} \, \mathrm{m}^{2}\]\[F = 0.54 \, \mathrm{N}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

pressure conversion
To solve problems involving different units of pressure, you often need to convert between units. In this example, the tire's internal pressure is given in bars, but we need it in Pascals (Pa), the standard SI unit for pressure. Converting units helps in maintaining consistency and ease in calculations.
unit conversion
Converting units is a crucial part of solving physics problems accurately. Here, we need to convert the area from \(\text{mm}^{2}\) to \( \text{m}^{2} \), the SI unit for area. One millimeter is 0.001 meters, so to convert \( \text{mm}^{2} \) to \( \text{m}^{2} \), you need to square this conversion factor.
force and area relationship
In physics, the force exerted by pressure on an area is given by the formula \( F = P \times A \). This relationship helps illustrate how pressure affects objects. If you know the pressure (P) acting on a surface of known area (A), you can find the force (F) acting on that surface. Understanding this relationship is key to solving many physics problems involving pressure.
SI units
The International System of Units (SI) is the globally accepted standard for measurements. Using SI units ensures consistency in scientific calculations. Common SI units include meters (m) for length, kilograms (kg) for mass, seconds (s) for time, and Pascals (Pa) for pressure.

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Most popular questions from this chapter

A \(1-\mathrm{L}\) tank holds \(1 \mathrm{g}\) of hydrogen at \(0^{\circ} \mathrm{C}\) and another 1-L tank holds 1 g of oxygen at \(0^{\circ}\) C. The mass of an oxygen molecule is 16 times the mass of a hydrogen molecule. (a) Do the tanks hold the same number of molecules? If not, which holds more? (b) Do the gases exert the same pressure? If not, which exerts the greater pressure? (c) Do the molecules in the tanks have the same average energy? If not, in which are the average energies greater? (d) Do the molecules in the tanks have the same average speeds? If not, in which are the average speeds greater?

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