Measurement, Units, and Uncertainty in Physics

Measurement, Units, and Uncertainty in Physics

Introduction to Measurement in Physics

Physics relies on precise measurement of physical quantities. To ensure consistency and clarity, scientists use standardized units for all measurements. The International System of Units (SI) is the globally accepted metric system for scientific work.

• Physical Quantity: Any property of matter or energy that can be measured (e.g., length, mass, time).

• Unit: A standard reference used to express the measurement of a physical quantity.

SI Base Units

There are seven SI base quantities, each with a corresponding SI unit. In introductory physics, three are most commonly used:

• Length: meter (m)

• Mass: kilogram (kg)

• Time: second (s)

These units form the foundation for all other measurements in physics.

Derived Units

Derived units are combinations of base units used to express other physical quantities.

• Density: $\text{kg/m}^3$

• Energy: $\text{kg} \cdot \text{m}^2/\text{s}^2$ (Joule, J)

Derived units are written as products or quotients of base units.

Unit Conversions and Conversion Factors

To convert between different units, conversion factors are used. A conversion factor is a ratio equal to 1, allowing units to be changed without altering the value of a quantity.

• Example conversion factors:

  • $1 \text{ min} = 60 \text{ s}$

  • $1 \text{ mile} = 5280 \text{ ft}$

  • $1 \text{ kg} = 2.2046 \text{ lbs}$

• Example problem: Convert $93 \text{ lbs} \cdot \text{ft}^2/\text{min}^2$ to $\text{kg} \cdot \text{m}^2/\text{s}^2$.

  • Apply conversion factors stepwise to cancel units:

  • $93 \frac{\text{lbs} \cdot \text{ft}^2}{\text{min}^2} \times \frac{1 \text{ kg}}{2.2046 \text{ lbs}} \times \left(\frac{1 \text{ m}}{3.2808 \text{ ft}}\right)^2 \times \left(\frac{1 \text{ min}}{60 \text{ s}}\right)^2$

  • Result: $0.0011 \text{ kg} \cdot \text{m}^2/\text{s}^2$

Measurements and Uncertainty

All measured numbers are inexact and have some degree of uncertainty. It is important to express this uncertainty clearly.

• Uncertainty: The range within which the true value is expected to lie, often expressed as $\pm$ a value.

• Significant Figures: Digits in a measurement that are known with certainty plus one estimated digit.

• Example: 2.5 cm has two significant figures; the last digit (5) is uncertain: $2.5 \pm 0.1$ cm.

Reporting Measurements and Precision

Measurements should always be reported to one digit beyond the smallest marked precision of the measuring tool.

• Example: If a graduated cylinder is marked every 1 mL, report the volume as 32.5 mL, not just 32 mL.

• Visual Example: A ruler marked in centimeters allows measurement to the nearest 0.1 cm.

Rules for Significant Figures

• All nonzero digits are significant. (e.g., 144 g has 3 sig figs)

• Zeros between nonzero digits are significant. (e.g., 8.0703 mL has 5 sig figs)

• Leading zeros are not significant. (e.g., 0.041 g has 2 sig figs)

• Trailing zeros to the right of a decimal point are significant. (e.g., 0.800 g has 3 sig figs)

• Trailing zeros in a whole number without a decimal point are ambiguous. Use scientific notation to clarify.

Calculations with Significant Figures

• Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

• Multiplication/Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Example: $12.025 \times 0.64 = 7.7$ (rounded to 2 sig figs)

Summary Table: SI Base Units

Physical Quantity	SI Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

Example: Reporting Measurement with Uncertainty

• Suppose a digital memory card is measured to be 2.9 cm wide using a ruler marked in millimeters. The measurement should be reported as 2.90 cm to reflect the precision of the tool.

• If a graduated cylinder reads 18.44 mL, the uncertainty is typically $\pm 0.01$ mL.

Additional info: Understanding units, conversions, and uncertainty is foundational for all further study in physics, as it ensures clarity and accuracy in scientific communication and problem-solving.