Physics: Derived Units and Systems

Derived Units

• Derived units are units for quantities derived from fundamental quantities.
• Examples include m/s for speed, g/cm³ for density.

Writing Units

• Unit symbols for units named after people are capitalized.
• Other unit symbols should not be capitalized.
• Symbols for derived units formed by multiplication are joined with a center dot or non-breaking space (e.g., N⋅m or Nm).
• Symbols for derived units formed by division are joined with a solidus (/) or negative exponent (e.g., m/s or ms⁻¹).
• Symbols for units are not pluralized (e.g., 30 m, 5 erg, not 30 ms or 5 ergs).

Systems of Units

• F.P.S. (Foot-Pound-Second): An older system used in Britain. Length is in feet, mass in pounds, and time in seconds.
• C.G.S. (Centimeter-Gram-Second): Length in centimeters, mass in grams, and time in seconds.
• M.K.S. (Meter-Kilogram-Second): Length in meters, mass in kilograms, and time in seconds.

Introduction to Physics

• Physics deals with matter and energy, and their interactions.
• It encompasses mechanics, optics, heat, electricity, and other branches.
• Mathematics is a powerful tool in physics, and new discoveries often arise from their combination.

Mechanics

• Mechanics describes bodies in motion or at rest.
• Newtonian mechanics, focusing on the behavior of objects under the influence of forces, is a critical part of physics.

Subcategories of Mechanics

• Statics: Deals with bodies at rest.
• Kinetics: Deals with bodies in motion:
  • Kinematics: Describes how bodies move without considering the cause of motion.
  • Dynamics: Describes how bodies move taking into account the cause of motion.

Physical Quantities

• A physical quantity is a measurable quantity.
• Basic quantities include mass, length, and time.

Types of Quantities

• Fundamental Quantities: Quantities that do not depend on other quantities, e.g., mass, time, length.
• Derived Quantities: Quantities derived from fundamental quantities, e.g., density, speed.

Particle vs. Body

• A particle or point object has no size.
• A body has size, mass, and volume.

Measurable Quantity

• Physical quantities have both a magnitude and a unit.
• Magnitude: Numerical value representing the quantity.
• Unit: Standard measure to express the quantity (e.g., meters, kilograms, seconds).

Types of Physical Quantities

• Fundamental (Basic) Quantity: Quantities that do not depend on other quantities. There are seven fundamental quantities in physics:
• Length
• Mass
• Time
• Illuminating power
• Temperature (Heat)
• Electric current
• Amount of chemical substance (Nuclear Physics)
• Plane angle
• Solid angle
• Derived Quantity: Quantities obtained from fundamental quantities, e.g.:
• Area: Length × Breadth
• Density: Mass/Length
• Speed: Distance/Time

Measurement in Physics

• Physics is a quantitative science; therefore, measurement is crucial.
• Measurement is the process of comparing an unknown quantity with a known standard unit.
• The basic principle of measurement is that the product of magnitude and units remains constant for a measurement, regardless of the system of units.

Unit of a Physical Quantity

• A unit is a reference standard for measurement.
• Units are of two kinds:
• Fundamental Unit: The unit of a fundamental physical quantity.
• Derived Unit: Units derived from fundamental units.

System of Units

• A complete set of units is called a system of units.
• F.P.S. System (Foot-Pound-Second):
• Length: foot
• Mass: pound (lb)
• Time: second (sec)
• C.G.S. System (Centimeter-Gram-Second):
• Length: centimeter
• Mass: gram
• Time: second
• M.K.S. System (Meter-Kilogram-Second):
• Length: meter
• Mass: kilogram
• Time: second

International System of Units (SI)

• Used in science and technology.

Basic SI Units

• Length: meter (m)
• Mass : kilogram (kg)
• Time : second (s)
• Electric Current: ampere (A)
• Temperature: kelvin (K)
• Luminous Intensity: candela (Cd)
• Amount of chemical substance: mole (mol)

Sub-fundamental SI Units

• Plane Angle: radian (rad)
• Solid Angle: steradian (sr)

Plane Angle

• The inclination between two lines.
• It is a 2-dimensional angle.

Solid Angle

• The angle made by an area cut on a sphere at its center.
• It is a 3-dimensional angle.

Advantages of SI Units

• Coherent System: Derived units are calculated by multiplication or division of fundamental units.
• Rationalized System: Only one unit is used for a given physical property (e.g., energy types using the joule).
• Metric System: Multiples and submultiples are expressed using powers of 10.
• Comprehensive: The seven SI units cover all scientific disciplines.
• Reproducible: SI units can be consistently recreated.
• Invariant in Time: The units remain constant over time.

Non-Standard Units

• Examples: Firkin (unit of volume).

Hybrid Units

• Combinations of units from different systems (e.g., ohm-centimeter).

Useful Non-SI Units

• Light Year: Distance light travels in one year. (9.5 x 10^15 m)
• Parsec: Unit of distance. (3.26 light years)
• Fermi/Femtometer: 10^-15 m
• Angstrom: 10^-10 m
• Astronomical Unit (AU): Distance between Earth and the Sun. (1.5 x 10^11 m)
• League: Unit of length, equivalent to 3 miles. (4828.03 m)
• Mile: 1.609 km
• Liter: 10^-3 m^3
• Gallon: 3.788 liters

Dimensions of a Physical Quantity

• All physical quantities can be expressed using fundamental quantities (e.g., length, mass, time).
• The powers of the fundamental quantities define the dimensions of a physical quantity.

Dimensional Analysis

• Basic Quantities and their Dimensions: | Quantity | Symbol | Dimension | |---|---|---| | Length | L | [L] | | Mass | M | [M] | | Time | T | [T] | | Temperature | | [K] or [Θ] | | Electric Current | I | [A] or [I] | | Amount of substance | | [N] | | Luminous Intensity | | [J] |

Mechanics (Derived Quantities)

• Speed: [LT⁻¹]
• Acceleration: [LT⁻²]
• Force: [MLT⁻²]
• Area: [L²]

Additional Notes on Dimensions

• Trigonometric functions and constants are dimensionless.
• Dimensionless quantities: Ratios of similar quantities are dimensionless (e.g., angle, trigonometric functions, specific gravity).
• Constant quantities: Constants may or may not have dimensions. Dimensionality shows the nature of a quantity, not whether it's constant.
• Dimensional notation conventions: Square brackets [ ] denote the dimensional formula of a quantity. Other bracket notations are sometimes used, but square brackets are the standard.
• Arbitrary Basic Quantities: Basic quantities can be chosen in arbitrary ways with corresponding units. Using a different set of units can yield different dimensional formulas for the same quantity.

Dimensional Formulae

• Length can be expressed in terms of force (F), acceleration (A), and time (T) with the formula: [L] = [F]^0 [A]^1 [T]^2, which simplifies to [L] = [A]^1 [T]^2.
• This means that length is directly proportional to acceleration and the square of time.
• Mass can be expressed in terms of force (F), velocity (v), and time (T) with the formula [M] = [F]^1 [v]^1 [T]^1, which simplifies to [M] = [F][v][T].
• This means that mass is directly proportional to the force, velocity, and time.

Principle of Homogeneity of Dimensions

• Dimensions are the fundamental units used to measure a physical quantity.
• Homogeneity of Dimensions states that each term within a physical equation must have the same dimensions.
• Example: The equation 6m = 6kg is physically incorrect because meters and kilograms have different dimensions.

Main Uses of Dimensional Analysis

• Checking the correctness of physical equations: Dimensional analysis can verify if the units in a physical equation are consistent.
• Deduce physical relationships: By analyzing the dimensions of involved quantities, we can find potential relationships between them.
• To convert units: Dimensional analysis can help convert units from one system to another.

Self Test Opportunity 1.1

• Density can be expressed in terms of force, length, and time with the formula [ρ] = [F]^1 [L]^-3 [T]^-2, which simplifies to [ρ] = [F][L]^-3 [T]^-2.
• This means that density is directly proportional to force and inversely proportional to length cubed and the square of time.
• Energy can be expressed in terms of force, length, and time with the formula [Energy] = [F]^1 [L]^1 [T]^2, which simplifies to [Energy] = [F][L][T]^2.
• This means energy is directly proportional to the force, length, and the square of time.

This quiz focuses on derived units in physics, including their symbols, capitalization rules, and examples like speed and density. Additionally, it covers various systems of units such as F.P.S., C.G.S., and M.K.S. Test your understanding of these fundamental concepts in physics.