Fundamental and Derived Quantities in Physics

• Physics is a quantitative science based on measurement of physical quantities.
• A physical quantity is a quantity that can be measured and described by a number.
• Units are standards for measuring physical quantities.

Fundamental and Derived Quantities

• Fundamental quantities are independent and not defined in terms of other quantities (e.g., length, mass, time).
• Derived quantities are defined in terms of fundamental quantities (e.g., speed, force, energy).
• Examples of fundamental quantities: length (L), mass (M), time (T), electric current (I), thermodynamic temperature (Θ), amount of substance (N), and luminous intensity (J).
• Examples of derived quantities: area, volume, density, velocity, acceleration, force, energy, power, etc.

Systems of Units

• A system of units is a complete set of units, both fundamental and derived, for all kinds of physical quantities.
• Common systems of units include:
  • CGS (centimeter, gram, second)
  • FPS (foot, pound, second)
  • MKS (meter, kilogram, second)
  • SI (International System of Units)

SI Units

• SI units are the internationally accepted system of units.
• There are seven base SI units:
  • Length: meter (m)
  • Mass: kilogram (kg)
  • Time: second (s)
  • Electric current: ampere (A)
  • Thermodynamic temperature: kelvin (K)
  • Amount of substance: mole (mol)
  • Luminous intensity: candela (cd)
• SI prefixes are used to denote multiples and submultiples of SI units (e.g., kilo, mega, giga, milli, micro).
• Examples of SI derived units:
  • Area: square meter (m²)
  • Volume: cubic meter (m³)
  • Density: kilogram per cubic meter (kg/m³)
  • Velocity: meter per second (m/s)
  • Acceleration: meter per second squared (m/s²)
  • Force: newton (N), where 1 N = 1 kg⋅m/s²
  • Energy: joule (J), where 1 J = 1 N⋅m = 1 kg⋅m²/s²
  • Power: watt (W), where 1 W = 1 J/s = 1 kg⋅m²/s³

Dimensions of Physical Quantities

• Dimensions represent the qualitative nature of a physical quantity.
• Dimensions are denoted by square brackets [ ].
• The dimensions of fundamental quantities are:
  • Length: [L]
  • Mass: [M]
  • Time: [T]
  • Electric Current: [I]
  • Temperature: [Θ]
  • Amount of Substance: [N]
  • Luminous Intensity: [J]
• The dimensions of derived quantities are expressed in terms of the dimensions of the fundamental quantities.
• Examples of dimensional formulas:
  • Area: [L²]
  • Volume: [L³]
  • Density: [ML⁻³]
  • Velocity: [LT⁻¹]
  • Acceleration: [LT⁻²]
  • Force: [MLT⁻²]
  • Energy: [ML²T⁻²]
  • Power: [ML²T⁻³]

Dimensional Analysis

• Dimensional analysis is a technique used to check the correctness of equations and to derive relationships between physical quantities.
• Principle of homogeneity: An equation is dimensionally correct if the dimensions of each term on both sides of the equation are the same.
• Applications of dimensional analysis:
  • Checking the correctness of an equation
  • Deriving relationships between physical quantities
  • Converting units from one system to another
• Limitations of dimensional analysis:
  • It cannot determine dimensionless constants in a formula.
  • It cannot be used to derive relations involving trigonometric, exponential, or logarithmic functions.
  • It does not tell whether a physical quantity is a vector or a scalar.

Significant Figures and Error Analysis

• Significant figures are the digits in a number that are known with certainty plus one uncertain digit.
• Rules for determining significant figures:
  • All non-zero digits are significant.
  • Zeros between non-zero digits are significant.
  • Leading zeros are not significant.
  • Trailing zeros in a number containing a decimal point are significant.
  • Trailing zeros in a number not containing a decimal point may or may not be significant.
• Rules for arithmetic operations with significant figures:
  • Addition and subtraction: The result should have the same number of decimal places as the number with the fewest decimal places.
  • Multiplication and division: The result should have the same number of significant figures as the number with the fewest significant figures.
• Error analysis involves estimating the uncertainty in measurements.
• Types of errors:
  • Systematic errors: Errors that are consistent and repeatable (e.g., zero error in an instrument).
  • Random errors: Errors that are unpredictable and vary from measurement to measurement.
• Absolute error is the difference between the measured value and the true value.
• Relative error is the absolute error divided by the true value.
• Percentage error is the relative error expressed as a percentage.
• Combination of errors:
  • Errors in addition and subtraction: The absolute errors add up.
  • Errors in multiplication and division: The relative errors add up.
  • Error in power: If z = xⁿ, then Δz/z = n(Δx/x).

Accuracy and Precision

• Accuracy refers to how close a measurement is to the true value.
• Precision refers to how close a set of measurements are to each other.
• It is possible to have precise measurements that are not accurate, and vice versa.

Examples of Dimensional Analysis

• To check the correctness of the equation v = u + at:
  • [v] = [LT⁻¹]
  • [u] = [LT⁻¹]
  • [at] = [LT⁻²][T] = [LT⁻¹]
• Since the dimensions of all terms are the same, the equation is dimensionally correct.
• To derive the relationship between time period (T), length (L), and acceleration due to gravity (g) for a simple pendulum:
  • Assume T = kLᵃgᵇ, where k is a dimensionless constant.
  • [T] = [L]ᵃ[LT⁻²]ᵇ
  • [T] = [Lᵃ⁺ᵇT⁻²ᵇ]
  • Comparing the powers, we get: a + b = 0 and -2b = 1
  • Solving these equations, we get: b = -1/2 and a = 1/2
  • Therefore, T = k√(L/g)