Summary: Physical Quantities And Units

Table of Contents
1. Introduction
2. What are Physical Quantities?
3. Seven Fundamental Physical Quantities
4. Fundamental and Derived Quantities
5. System of Units
6. Rules for Symbols and S.I. Prefixes
7. Order of Magnitude
8. Classification of Physical Quantities
9. Dimensions of a Physical Quantity
10. Dimensional Analysis
11. Significant Figures and Rounding Off
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Introduction

• When building or measuring, different teams must use the same units so parts fit and structures remain safe.
• Accurate measurements and consistent units prevent costly errors and repairs.

What are Physical Quantities?

• A physical quantity is something that can be measured and described by physics; it has two parts: a numerical magnitude and a unit.
• Expressed as n × u (numerical value × unit). If the unit changes, the numerical magnitude changes so that the product remains the same.
• Larger the unit, smaller will be the magnitude.

Seven Fundamental Physical Quantities

• Length
• Mass
• Time
• Electric current
• Temperature
• Amount of substance
• Luminous intensity

Fundamental and Derived Quantities

• Fundamental quantities are independent and serve as bases for others.
• Derived quantities are formed by multiplying/dividing powers of fundamental quantities (e.g., area = length², volume = length³).

System of Units

• A full set of fundamental and derived units is called a system of units.
• Common systems: CGS (cm, g, s), MKS (m, kg, s), FPS (ft, lb, s), and the global standard S.I..
• Two supplementary units: radian (rad) for plane angle and steradian (sr) for solid angle.

Rules for Symbols and S.I. Prefixes

• Unit symbols are written in Roman (upright) type, not italics.
• Unit names are not capitalized even if named after a person; their symbol may be a capital letter when named after a scientist.
• Symbols use the singular form and have no final full stop; plurals are not used.
• Slash (/) is used only to indicate division of one unit symbol by another; at most one slash is used.
• Prefixes are written in Roman type, placed immediately before the unit symbol (no space) and form a single combined symbol that can be raised to powers.
• Prefixes are never used alone; double prefixes are avoided when a single prefix exists.

Order of Magnitude

• Order of magnitude indicates the largeness or smallness of a quantity.
• Express the value as a × 10^b with 1 ≤ a < 10; the exponent b is the order of magnitude.

Classification of Physical Quantities

• Based on direction: scalars (only magnitude) and vectors (magnitude and direction).
• Based on dependency:
  • Fundamental quantities: chosen independently to express others.
  • Derived quantities: expressed from base quantities and potentially infinite in number.
  • Supplementary quantities: plane angle and solid angle (have units but are dimensionless).

Dimensions of a Physical Quantity

• Dimensions are the powers of fundamental units used to express a derived quantity.
• Knowing a quantity's unit lets you write its dimensional formula in terms of base dimensions.

Dimensional Analysis

• Principle: only quantities with the same dimensions can be added/subtracted; equality requires same dimensions.
• Uses:
  • Check dimensional correctness of equations.
  • Help derive relations between physical quantities.
  • Convert between unit systems.
• Limitations:
  • Gives no information about dimensionless constants.
  • Cannot determine formulas if a quantity depends on more than three variables in the method used here.
  • Cannot find formulas involving trigonometric or exponential functions.
  • An equation can be dimensionally correct but still incorrect physically.

Significant Figures and Rounding Off

• Significant figures are measured digits plus one estimated digit.
• Rules:
  • All nonzero digits are significant.
  • Zeros between significant digits are significant.
  • Leading zeros are not significant.
  • Trailing zeros are significant only if to the right of a decimal point.
• Rounding rules:
  • If the dropped part is greater than 5 → round up; less than 5 → round down.
  • If exactly 5 with no following digits, use round-half-to-even rule.
  • For addition/subtraction: round to the least precise decimal place.
  • For multiplication/division: round to the least number of significant figures among inputs.