1.2 Physical Quantities and Units

Fundamental Units and Measurements

Physics depends on measurement, and measurements need units. The SI (Système International) provides a universal set of units so scientists everywhere can compare results without confusion. This section covers the core SI units, how to convert between measurement systems, and how to handle the very large and very small numbers that show up constantly in physics.

Conversion of Measurement Systems

Two major systems of units exist side by side: the SI (metric) system used in science worldwide, and the US customary (English) system still common in everyday American life. Knowing how to convert between them is a practical skill you'll use throughout this course.

• Length
  • SI unit: meter (m)
  • US customary unit: foot (ft)
  • Conversion: $1 \text{ m} = 3.281 \text{ ft}$
• Mass
  • SI unit: kilogram (kg)
  • US customary unit: pound (lb)
  • Conversion: $1 \text{ kg} = 2.205 \text{ lb}$
  • Note: technically, the pound is a unit of force in the US system, but it's commonly used for mass in everyday contexts. In physics, be careful to distinguish mass (kg) from weight (N).
• Time
  • SI unit: second (s)
  • The second is the same in both systems, so no conversion is needed.
• Temperature
  • SI unit: kelvin (K)
  • US customary unit: degree Fahrenheit (°F)
  • Conversion: $T(K) = \frac{T(°F) + 459.67}{1.8}$
  • The Celsius scale (°C) is also metric and often used in science. To go from Celsius to kelvin: $T(K) = T(°C) + 273.15$
• Force
  • SI unit: newton (N), defined as the force needed to accelerate 1 kg at $1 \text{ m/s}^2$
  • US customary unit: pound-force (lbf)
  • Conversion: $1 \text{ N} = 0.2248 \text{ lbf}$

Expression of Extreme Measurements

Physics deals with quantities ranging from the size of a proton to the distance between galaxies. Metric prefixes, scientific notation, and significant figures are the tools that make these numbers manageable.

Metric Prefixes

Metric prefixes attach to a base unit to scale it up or down by powers of 10. You should memorize the most common ones:

Scientific notation expresses a number as a value between 1 and 10 multiplied by a power of 10. This keeps very large and very small numbers compact and readable.

• Avogadro's number: $6.02 \times 10^{23}$ (number of particles in one mole)
• Diameter of a hydrogen atom: roughly $1.06 \times 10^{-10}$ m
• Distance from Earth to the Sun: about $1.50 \times 10^{11}$ m

To write a number in scientific notation:

1. Move the decimal point until you have a number between 1 and 10.
2. Count how many places you moved it. That count becomes the exponent on 10.
3. If you moved the decimal to the left, the exponent is positive. If you moved it to the right, the exponent is negative.

Significant Figures

Significant figures tell you how precise a measurement is. Every measuring tool has a limit, and significant figures reflect that limit honestly.

• All nonzero digits are significant: 245 has 3 significant figures.
• Zeros between nonzero digits are significant: 1,003 has 4 significant figures.
• Leading zeros are not significant: 0.0042 has 2 significant figures.
• Trailing zeros after a decimal point are significant: 2.50 has 3 significant figures.

When multiplying or dividing, your answer should have the same number of significant figures as the measurement with the fewest significant figures. When adding or subtracting, round to the least precise decimal place.

Fundamental SI Units

The SI system is built on seven base units. For this course, the most important ones are:

• Meter (m) for length: defined as the distance light travels in a vacuum in $\frac{1}{299,792,458}$ of a second. This definition ties length to the speed of light, which is a universal constant.
• Kilogram (kg) for mass: as of 2019, the kilogram is defined using the Planck constant ($h = 6.626 \times 10^{-34} \text{ J·s}$), not a physical artifact. The old platinum-iridium cylinder in France served as the standard for over a century, but it could change mass over time through contamination or wear. The new definition is more stable and reproducible.
• Second (s) for time: defined as the duration of 9,192,631,770 oscillations of radiation from a cesium-133 atom. Atomic clocks based on this definition are accurate to about 1 second in 300 million years.
• Ampere (A) for electric current: since 2019, the ampere is defined by fixing the value of the elementary charge $e = 1.602 \times 10^{-19}$ coulombs. The older definition involving force between parallel wires was a thought experiment that couldn't be realized with high precision.

Measurement and Analysis

Base Units vs. Derived Units

Base units (meter, kilogram, second, ampere, kelvin, mole, candela) are the building blocks. Derived units are combinations of base units that describe more complex quantities. For example:

• The newton (N) is $\text{kg} \cdot \text{m/s}^2$
• The joule (J) is $\text{kg} \cdot \text{m}^2/\text{s}^2$
• The watt (W) is $\text{J/s}$, or $\text{kg} \cdot \text{m}^2/\text{s}^3$

Dimensional Analysis

Dimensional analysis is a technique for checking whether an equation makes physical sense and for converting units. The core idea: you can only add or equate quantities that have the same dimensions.

Here's how to use it for a unit conversion:

1. Write down the quantity you want to convert.
2. Multiply by a conversion factor written as a fraction (with the unit you want to cancel in the denominator).
3. Cancel units and compute.

For example, convert 5.0 km to meters:

$5.0 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 5000 \text{ m}$

You can also use dimensional analysis to verify equations. If you're told that distance equals velocity times time, check the dimensions: $\text{m/s} \times \text{s} = \text{m}$. The units work out, so the equation is at least dimensionally consistent.

Precision vs. Accuracy

These two words mean different things in physics:

• Precision is how close repeated measurements are to each other. High precision means low random error.
• Accuracy is how close a measurement is to the true value. Low accuracy usually points to systematic error (like a miscalibrated instrument).

Both matter when evaluating experimental results. Ideally, you want measurements that are both precise and accurate.