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Units & Measurement

Physics is a quantitative science — every result requires both a number and a unit. The International System of Units (SI) provides a universal framework so scientists worldwide can communicate unambiguously. Mastering dimensional analysis and significant figures is the foundation of all physics problem-solving.

Key Concepts

SI Base Units

Seven fundamental units from which all others are derived: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).

Dimensional Analysis

A method of checking equations and converting units by treating dimensions — length

L

, mass

M

, time

T

— as algebraic quantities that must balance on both sides of an equation.

Significant Figures

The meaningful digits in a measured quantity reflecting its precision. For multiplication/division, keep the fewest sig figs of any input. For addition/subtraction, keep the fewest decimal places.

Scientific Notation

Writing numbers as

a \times 10^n

where

1 \leq a < 10

. For example, the speed of light is

c = 2.998 \times 10^8 \text{ m/s}

Accuracy vs. Precision

Accuracy measures closeness to the true value. Precision measures reproducibility — how close repeated measurements are to each other. A measurement can be precise without being accurate.

Measurement Uncertainty

Every measurement has inherent uncertainty

\delta x

. We write results as

x \pm \delta x

. The relative uncertainty is

\delta x / x

, often expressed as a percentage.

Key Equations

Unit Conversion (identity fraction)

\frac{1 \text{ m}}{100 \text{ cm}} = 1

Multiplying any quantity by a unit-conversion fraction equal to 1 changes the unit without changing the physical quantity.

Dimensions of Common Quantities

[v] = \frac{L}{T}, \quad [a] = \frac{L}{T^2}, \quad [F] = \frac{ML}{T^2}

Dimensions of velocity, acceleration, and force in terms of base dimensions L, M, T.

Relative Uncertainty

\text{relative uncertainty} = \frac{\delta x}{x} \times 100\%

Expresses measurement precision as a percentage of the measured value.

Uncertainty Propagation (product/quotient)

\frac{\delta(xy)}{xy} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2}

For a product or quotient, relative uncertainties add in quadrature.

Worked Example

Unit Conversion: mph to m/s

Problem

Convert a speed of 60 miles per hour to meters per second.

Solution

Write the quantity with its unit.

60 \, \frac{\text{mi}}{\text{hr}}

Multiply by conversion fractions equal to 1 (1 mi = 1609 m, 1 hr = 3600 s):

60 \, \frac{\text{mi}}{\text{hr}} \times \frac{1609 \text{ m}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}}

Cancel units and compute:

= \frac{60 \times 1609}{3600} \, \frac{\text{m}}{\text{s}} \approx 26.8 \, \text{m/s}

Answer 60 mph ≈ 26.8 m/s

Practice

Exercises

7 problems

1 of 7

Convert $5.0 \text{ km}$ to meters.

Your answer

2 of 7

Convert $90 \text{ km/h}$ to meters per second.

Your answer

m/s

3 of 7

Convert $2.5$ hours to seconds.

Your answer

4 of 7

A measurement is recorded as $8.0 \pm 0.04$ m. What is the percentage uncertainty?

Your answer

5 of 7

Convert $1.50 \times 10^3$ mm to meters.

Your answer

6 of 7

A rectangular box measures $0.50 \text{ m} \times 0.40 \text{ m} \times 0.20 \text{ m}$ . What is its volume?

Your answer

m³

7 of 7

The speed of sound in air is $343 \text{ m/s}$ . Express this speed in km/h.

Your answer

km/h

Key Takeaways

The seven SI base units underpin all physical measurement.
Dimensional analysis checks equations and guides unit conversions — dimensions must match on both sides.
Significant figures propagate through calculations: multiply/divide → fewest sig figs; add/subtract → fewest decimal places.
State every measurement result as $x \pm \delta x$ to communicate both value and uncertainty.
Always sanity-check numerical answers using dimensional analysis before accepting them.