Question

Appendix B lists multiple units along with whether or not they are SI base units. The joule, volt, and pascal are all SI derived units. Express each of these units in terms of SI base units.

Short answer

Joule: \( \text{kg} \cdot \text{m}^2/\text{s}^2 \); Volt: \( \text{kg} \cdot \text{m}^2/(\text{A} \cdot \text{s}^3) \); Pascal: \( \text{kg}/(\text{m} \cdot \text{s}^2) \).

Step-by-step solution

Understanding SI Base Units

SI Base Units are the fundamental units in the International System of Units, from which all other units are derived. The seven SI base units are: meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for the amount of substance, and candela (cd) for luminous intensity.

Identify Derived Unit Equations

The derived units joule, volt, and pascal are combinations of base units. A joule is a unit of energy, a volt is a unit of electric potential, and a pascal is a unit of pressure. These need to be expressed in terms of SI base units.

Expressing Joule in SI Base Units

A joule is defined as the amount of work done when a force of one newton is applied over a distance of one meter. Since force is expressed in newtons and a newton is derived as \[ \text{N} = \text{kg} \cdot \text{m}/\text{s}^2, \]a joule can be expressed as:\[ \text{J} = \text{N} \cdot \text{m} = (\text{kg} \cdot \text{m}/\text{s}^2) \cdot \text{m} = \text{kg} \cdot \text{m}^2/\text{s}^2. \]

Expressing Volt in SI Base Units

A volt is the potential difference that will move one coulomb of charge with one joule of energy. It can also be defined as 1 joule per coulomb. Since coulomb can be expressed using ampere and time, a volt is:\[ \text{V} = \text{W}/\text{C} = \text{J}/(\text{A} \cdot \text{s}) = (\text{kg} \cdot \text{m}^2/\text{s}^2) / (\text{A} \cdot \text{s}) = \text{kg} \cdot \text{m}^2/(\text{A} \cdot \text{s}^3). \]

Expressing Pascal in SI Base Units

A pascal is defined as one newton per square meter. Since one newton is already expressed in SI base units, substituting it in gives:\[ \text{Pa} = \text{N}/\text{m}^2 = (\text{kg} \cdot \text{m}/\text{s}^2) / \text{m}^2 = \text{kg}/(\text{m} \cdot \text{s}^2). \]

Key concepts

Joule in SI Units

The joule, symbolized as \( ext{J} \), is a derived SI unit used to measure energy, work, or heat. Understanding its expression in terms of SI base units can make its application in physics and engineering more intuitive.

The joule is defined as the amount of energy transferred or work done when a force of one newton causes a displacement of one meter. A newton, the SI unit of force, is defined as \( ext{kg} \cdot ext{m}/ ext{s}^2 \).

Therefore, the joule can be broken down into these base units:

• Joule \( ( ext{J} ) \)
• \( ext{J} = ext{N} \cdot ext{m} \)
• \( ext{J} = ( ext{kg} \cdot ext{m}/ ext{s}^2) \cdot ext{m} \)
• \( ext{J} = ext{kg} \cdot ext{m}^2/ ext{s}^2 \)

These equations show the joule as a combination of the kilogram, meter, and second, providing a direct link to fundamental physical concepts.

Volt in SI Units

In the realm of electricity, the volt, denoted by \( ext{V} \), is an essential unit that measures electric potential difference, electric pressure, or electromotive force.

A volt is defined in terms of the energy required to move charge. Specifically, 1 volt is equivalent to a potential difference that moves 1 coulomb of charge with an energy of 1 joule. With this definition, a volt connects energy and electrical charge.

Breaking it down into base units involves understanding a coulomb, which can be expressed using the ampere and time. Therefore:

• Volt \( ( ext{V} ) \)
• \( ext{V} = ext{J}/ ext{C} \)
• \( ext{V} = ( ext{kg} \cdot ext{m}^2/ ext{s}^2) / ( ext{A} \cdot ext{s}) \)
• \( ext{V} = ext{kg} \cdot ext{m}^2/( ext{A} \cdot ext{s}^3) \)

This breakdown into base units makes the volt relatable to basic quantities such as mass, length, time, and current, providing a clearer understanding of its role in physics.

Pascal in SI Units

The pascal (\( ext{Pa} \)) serves as the standard unit for pressure in the International System of Units. It measures force per unit area and is a vital concept in fluid dynamics and atmospheric sciences.

Defined as one newton per square meter, the pascal connects force and area in a straightforward manner. Since a newton is already expressed in basic units, the conversion of a pascal is directly derived.

Here’s how the pascal breaks down into base units:

• Pascal \( ( ext{Pa} ) \)
• \( ext{Pa} = ext{N}/ ext{m}^2 \)
• \( ext{Pa} = ( ext{kg} \cdot ext{m}/ ext{s}^2) / ext{m}^2 \)
• \( ext{Pa} = ext{kg}/( ext{m} \cdot ext{s}^2) \)

This detailed expression ties back to mass, length, and time, offering an engaging insight into the concept of pressure and its measurement.