### Halliday Solution: Fundamental of Physics 7th Edition Chapter Solutions
#### Knowledge Points:
1. **Conversion Between Units and Dimensions:**
- The text demonstrates the process of converting distances from furlongs to rods and chains, utilizing given conversion factors. For example, it calculates the distance in rods as:
\[
d = 4.0 \text{ furlongs} \times \frac{201.168 \text{ m}}{\text{furlong}} \times \frac{1 \text{ rod}}{5.0292 \text{ m}} = 160 \text{ rods}
\]
- Similarly, the distance is converted into chains:
\[
d = 4.0 \text{ furlongs} \times \frac{201.168 \text{ m}}{\text{furlong}} \times \frac{1 \text{ chain}}{20.117 \text{ m}} = 40 \text{ chains}
\]
- This section emphasizes the importance of understanding and applying conversion factors correctly to solve problems involving different units.
2. **Unit Conversions for Small Quantities:**
- The conversion of very small units like gry (a unit of length used in printing and typography) to points (another unit of length) is explained:
\[
1 \text{ gry} = \left(\frac{1}{10}\right) \left(\frac{1}{12}\right) \left(72 \text{ points}\right) = 0.60 \text{ point}
\]
- The square of this conversion is then calculated:
\[
1 \text{ gry}^2 = (0.60 \text{ point})^2 = 0.36 \text{ point}^2
\]
- This example highlights the precision required when dealing with minute measurements and the need for careful handling of units and their conversions.
3. **Metric Prefixes and Unit Conversion:**
- Metric prefixes (micro, pico, nano, etc.) are discussed, emphasizing their use in converting between units. For instance, the conversion from kilometers to micrometers:
\[
1 \text{ km} = 1 \times 10^3 \text{ m} = 1 \times 10^6 \mu \text{m}
\]
- The text also explains how to calculate the fraction of one centimeter equal to 1.0 µm:
\[
1 \text{ cm} = 10^{-2} \text{ m} = 10^{4} \mu \text{m}
\]
- These examples demonstrate the practical application of metric prefixes in scientific calculations.
4. **Conversion Between Imperial and Metric Units:**
- Conversions between inches, picas, and centimeters are shown, using the exact conversion factor of 1 inch = 2.54 cm:
\[
0.80 \text{ cm} = 0.80 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} \times \frac{6 \text{ picas}}{1 \text{ inch}} = 1.9 \text{ picas}
\]
- Further conversion to points (12 points = 1 pica) is provided:
\[
0.80 \text{ cm} = 0.80 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} \times \frac{6 \text{ picas}}{1 \text{ inch}} \times \frac{12 \text{ points}}{1 \text{ pica}} = 23 \text{ points}
\]
- This section reinforces the ability to convert between different systems of measurement.
5. **Geometric Formulas Applied to Real-World Objects:**
- The circumference, surface area, and volume of the Earth are calculated using the radius \( R = 6.37 \times 10^6 \text{ m} \):
\[
\text{Circumference} = 2\pi R = 4.00 \times 10^4 \text{ km}
\]
\[
\text{Surface Area} = 4\pi R^2 = 5.10 \times 10^8 \text{ km}^2
\]
\[
\text{Volume} = \frac{4}{3}\pi R^3 = 1.08 \times 10^{12} \text{ km}^3
\]
- These calculations showcase the application of geometric formulas to estimate physical dimensions of objects, particularly in astronomy and geography.
6. **Historical Units of Measurement:**
- The conversion between historical units such as "fanega" and "cahiz" is explained, using Table 1-6 as a reference:
\[
1 \text{ fanega} = \frac{1}{12} \text{ cahiz} = 8.33 \times 10^{-2} \text{ cahiz}
\]
\[
1 \text{ cuartilla} = \frac{1}{48} \text{ cahiz} = 2.08 \times 10^{-2} \text{ cahiz}
\]
- This section provides insight into the historical context of units and their conversions, which can be useful in understanding cultural and regional differences in measurement systems.
These knowledge points cover various aspects of unit conversions, including the use of conversion factors, understanding of metric prefixes, and the application of geometric formulas to real-world scenarios. They emphasize the importance of accurate and precise calculations in scientific and engineering contexts.
