Physical Quantities, Units, and Measurement in Physics

Physical Quantities and Units

Definition of Physical Quantity

In physics, a physical quantity is any property of a phenomenon, body, or substance that can be quantified through measurement. Examples include length, mass, time, temperature, and electric current.

• Quantification: Physical quantities are measured and expressed as a number multiplied by a unit.

• Units: Numbers alone are meaningless without specifying the unit of measurement.

• System of Units: Units are defined relative to a system, most commonly the International System of Units (SI), which is a modern form of the metric system.

Base and Derived Quantities

• Base Quantities: The SI system is founded on seven fundamental (base) physical quantities (e.g., length, mass, time).

• Derived Quantities: Other quantities are derived from these base quantities (e.g., speed, area, volume).

• Mechanics: In introductory mechanics, only three base quantities are typically needed: length, mass, and time.

Definitions of Base Units

Time

• Unit: Second (s)

• Modern Definition: Since 1967, the second is defined by the frequency of radiation corresponding to the transition between two hyperfine energy states of the cesium-133 atom.

Length

• Unit: Meter (m)

• Modern Definition: Since 1983, the meter is defined as the distance light travels in a vacuum in 1/299,792,458 seconds. This sets the speed of light in a vacuum as exactly 299,792,458 m/s.

Mass

• Unit: Kilogram (kg)

• Modern Definition: As of 2019, the kilogram is defined by fixing the value of Planck’s constant ($h$) in SI units ($kg ext{ m}^2/ ext{s}$), along with the definitions of the second and meter.

• Prefix: The kilogram is the only SI base unit with a prefix ("kilo").

More About Units

Dimensional Consistency

Equations in physics must be both numerically and dimensionally consistent. This means that the units on both sides of an equation must match.

• Example: Distance = Speed × Time

• Units: $m = (m/s) imes s = m$

• Rules:

  • Only add quantities with the same units.

  • Always multiply quantities with different units.

  • Units can be manipulated algebraically (e.g., $m^2$, $m/s$).

• Checking Answers: Carry units through calculations to verify the physical meaning of results.

Converting Units

Unit Conversion Process

It is best practice to convert all physical quantities to SI units before using them in equations.

• Do not round numbers until the final answer is obtained.

• Conversion Factors: Multiplying by conversion factors changes only the units, not the value.

• Example: Convert 55 mi/h to m/s: $55 ext{ mi/h} imes rac{1.609 ext{ km}}{1 ext{ mi}} imes rac{1000 ext{ m}}{1 ext{ km}} imes rac{1 ext{ h}}{3600 ext{ s}} = 24.6 ext{ m/s}$

• Powers of Units: When converting units raised to a power (e.g., area, volume), raise the conversion factor to that power. Example: $50.7 ext{ cm}^2 imes igg( rac{1 ext{ m}}{100 ext{ cm}}igg)^2 = 0.00507 ext{ m}^2$

Uncertainty and Significant Figures

Uncertainty (Error)

Uncertainty or error is the maximum difference between a measured value and the true value of a physical quantity.

• Expressed as: measured value ± uncertainty

• Fractional Error: $\text{Fractional error} = \frac{\text{uncertainty}}{\text{measured value}}$

• Percent Error: $\text{Percent error} = \text{Fractional error} \times 100\%$

• Example: If a resistor is labeled 47 Ω ± 10%, the true value is in the range 42.3 Ω to 51.7 Ω.

Significant Figures

Significant figures are the digits in a measured value that are known with certainty plus one estimated digit.

• Examples:

  • 0.25 has 2 significant figures

  • 1.025 has 4 significant figures

  • 0.0025 has 2 significant figures

• Rules for Calculations:

  • Multiplication/Division: Result has as many significant figures as the factor with the fewest significant figures.

  • Addition/Subtraction: Result is rounded to the least number of decimal places among the terms.

Scientific Notation

Scientific notation is used to express very large or very small numbers and to clearly display significant figures.

• Example: The distance from Earth to the Moon is about $3.84 \times 10^8$ m (3 significant figures).

• Example: $4.00 \times 10^{-7}$ m has 3 significant figures.

Additional info:

• These notes cover foundational concepts in measurement and units, which are essential for all areas of physics, especially introductory mechanics.