1.3 The Language of Physics: Physical Quantities and Units

Physical Quantities, Units, and Measurement

Physics describes the natural world through measurable quantities, each expressed in standardized units. The SI (Système International) system gives scientists a common language so that a "kilogram" means the same thing in every lab on Earth. This section covers the building blocks: fundamental and derived quantities, how to convert between units, how to handle measurement precision, and how to read the relationships that graphs reveal.

Physical Quantities and SI Units

Every measurement in physics traces back to seven fundamental quantities. These can't be broken down into simpler measurements. Think of them as the "atoms" of measurement.

Derived quantities are built by combining fundamental quantities through multiplication or division. You'll encounter these constantly throughout the course:

• Area ($m^2$): length × length. Describes a two-dimensional surface.
• Volume ($m^3$): length × length × length. The space a 3D object occupies.
• Density ($kg/m^3$): mass ÷ volume. Water has a density of about $1000 \, kg/m^3$; air is roughly $1.2 \, kg/m^3$.
• Speed ($m/s$): distance ÷ time. A car on the highway might travel at about $30 \, m/s$.
• Acceleration ($m/s^2$): change in velocity ÷ time. Gravity near Earth's surface is $9.8 \, m/s^2$.
• Force (N = $kg \cdot m/s^2$): mass × acceleration. The newton is named after Isaac Newton.
• Energy (J = $kg \cdot m^2/s^2$): capacity to do work. Lifting a 1 kg object by 1 m against gravity takes about $9.8 \, J$.
• Power (W = $kg \cdot m^2/s^3$): energy ÷ time. A 60 W light bulb converts 60 joules every second.
• Pressure (Pa = $kg/(m \cdot s^2)$): force ÷ area. Atmospheric pressure at sea level is about $101{,}325 \, Pa$.

Notice the pattern: every derived unit breaks down into some combination of meters, kilograms, and seconds (plus amperes, kelvins, etc. when needed). If you can trace a unit back to its fundamentals, you understand what it actually measures.

Unit Conversions and Scientific Notation

Metric prefixes scale base units by powers of 10. The ones you'll use most often:

Example: Convert $5.0 \, km$ to meters.

$5.0 \, km \times \frac{1000 \, m}{1 \, km} = 5000 \, m = 5.0 \times 10^3 \, m$

Example: Convert $15 \, m/s$ to $km/h$.

$15 \, \frac{m}{s} \times \frac{1 \, km}{1000 \, m} \times \frac{3600 \, s}{1 \, h} = 54 \, km/h$

Dimensional analysis also serves as an error check. If your final answer doesn't have the right units, something went wrong in the calculation.

Types of Physical Quantities

• Scalar quantities have magnitude only. Examples: mass ($5 \, kg$), temperature ($300 \, K$), energy ($200 \, J$). A scalar is fully described by a single number with a unit.
• Vector quantities have both magnitude and direction. Examples: velocity ($20 \, m/s$ north), force ($10 \, N$ downward), displacement ($3 \, m$ east). You can't fully describe a vector without specifying where it points.

This distinction matters because vectors follow different math rules than scalars. Two forces of $5 \, N$ don't always add to $10 \, N$; it depends on their directions. You'll work with vector addition extensively later in the course.

Significant Figures in Calculations

Every measurement has a limit to its precision, and significant figures (sig figs) communicate that limit. When you write $3.05 \, m$, you're saying you're confident in the "3" and the "0" and reasonably confident in the "5." The rules for identifying sig figs:

1. Non-zero digits are always significant. $247$ has 3 sig figs.
2. Zeros between non-zero digits are significant. $1.0204$ has 5 sig figs.
3. Leading zeros are never significant. $0.0012$ has 2 sig figs (only the 1 and 2 count).
4. Trailing zeros after a decimal point are significant. $1.00$ has 3 sig figs.
5. Trailing zeros without a decimal point are ambiguous. $100$ could be 1, 2, or 3 sig figs. Using scientific notation removes the ambiguity: $1.00 \times 10^2$ clearly has 3.

Sig Fig Rules for Calculations

• Addition/subtraction: Round the result to the fewest decimal places of any measurement used.
  • $1.2 + 3.45 = 4.65$, rounded to $4.7$ (one decimal place, matching $1.2$)
• Multiplication/division: Round the result to the fewest significant figures of any measurement used.
  • $2.5 \times 3.42 = 8.55$, rounded to $8.6$ (two sig figs, matching $2.5$)

Accuracy vs. Precision

These two terms sound similar but mean different things:

• Accuracy is how close a measurement is to the true or accepted value. A bathroom scale that reads $80.0 \, kg$ when you actually weigh $80.2 \, kg$ is quite accurate.
• Precision is how reproducible your measurements are. If you step on the scale five times and get $80.0 \, kg$ every time, it's very precise.

A measurement can be precise without being accurate (consistently wrong), or accurate without being precise (scattered around the true value). The goal is both.

Graphs of Physical Relationships

Graphs are one of the most powerful tools in physics. They let you see the mathematical relationship between two quantities at a glance, and different graph shapes correspond to different types of equations.

Linear graphs (straight lines) indicate a constant rate of change.

• The equation has the form $y = mx + b$.
• Slope ($m$) represents the rate of change. On a position vs. time graph with constant velocity, the slope is the velocity.
• y-intercept ($b$) represents the initial value when $x = 0$.
• Example: plotting position vs. time for an object moving at constant velocity produces a straight line.

Quadratic graphs (parabolas) appear when one variable depends on the square of another.

• The equation has the form $y = ax^2 + bx + c$.
• The vertex represents a maximum or minimum (like the peak height of a thrown ball).
• Example: position vs. time for an object under constant acceleration is parabolic.

Inverse graphs (hyperbolas) show that as one variable increases, the other decreases proportionally.

• The equation has the form $y = k/x$, so the product $xy = k$ stays constant.
• Example: Boyle's law ($PV = k$) at constant temperature. As pressure increases, volume decreases.

Logarithmic/exponential relationships sometimes appear on graphs with logarithmic scales, which compress large ranges of data. A logarithmic scale turns an exponential curve into a straight line, making it easier to analyze. The decibel scale for sound intensity is a common example.

Creating Proper Graphs

1. Identify the independent variable (what you control) and place it on the x-axis. The dependent variable (what you measure) goes on the y-axis.
2. Choose scales that spread the data across most of the graph area. Don't cram all your points into one corner.
3. Label each axis with the quantity name and its unit, e.g., "Time (s)" or "Position (m)."
4. Plot each data point carefully and consistently.
5. Draw a best-fit line or curve that follows the overall trend. This line doesn't need to pass through every point; it represents the general relationship.

Interpreting Graphs

• Identify the shape first (linear, parabolic, hyperbolic) to determine the type of mathematical relationship.
• Calculate the slope where relevant. For a straight line, pick two points far apart on the best-fit line (not necessarily data points) and use $\text{slope} = \frac{\Delta y}{\Delta x}$.
• Use the graph to interpolate (estimate values between data points) or extrapolate (estimate values beyond the measured range). Extrapolation is less reliable since you're assuming the trend continues.