Physical Quantities

Introduction to Physical Quantities

Physical quantities are measurable properties of matter or energy used to describe physical phenomena. They are classified into fundamental and derived quantities based on whether they can be expressed in terms of basic units or are formed by their combination.

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Classification of Physical Quantities

1. Fundamental Quantities:

   • Independent and cannot be derived from other quantities.
   • Examples:
     • Length ($L$)
     • Mass ($M$)
     • Time ($T$)
     • Electric Current ($I$)
     • Temperature ($\Theta$)
     • Luminous Intensity ($J$)
     • Amount of Substance ($N$)

2. Derived Quantities:

   • Formed by combining fundamental quantities.
   • Examples:
     • Velocity: $[L T^{-1}]$
     • Force: $[M L T^{-2}]$
     • Pressure: $[M L^{-1} T^{-2}]$

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System of Units

1. CGS System:

   • Length: Centimeter ($cm$)
   • Mass: Gram ($g$)
   • Time: Second ($s$)

2. MKS System:

   • Length: Meter ($m$)
   • Mass: Kilogram ($kg$)
   • Time: Second ($s$)

3. SI System:

   • Length: Meter ($m$)
   • Mass: Kilogram ($kg$)
   • Time: Second ($s$)
   • Electric Current: Ampere ($A$)
   • Temperature: Kelvin ($K$)
   • Luminous Intensity: Candela ($cd$)
   • Amount of Substance: Mole ($mol$)

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Scalars and Vectors

1. Scalar Quantities:

   • Have only magnitude.
   • Examples: Distance, Speed, Mass, Temperature, Energy.

2. Vector Quantities:

   • Have both magnitude and direction.
   • Examples: Displacement, Velocity, Force, Acceleration.

Property	Scalars	Vectors
Magnitude	Yes	Yes
Direction	No	Yes
Addition	Simple arithmetic	Vector addition rules

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Representing Vector Quantities

1. Graphical Representation:

   • Represented as arrows, where:
     • Length = Magnitude.
     • Arrowhead = Direction.

2. Vector Addition:

   • Triangle Law: If two vectors are represented by two sides of a triangle in sequence, their resultant is the third side of the triangle taken in reverse order.
   • Parallelogram Law: The resultant vector is the diagonal of a parallelogram formed by the two vectors.

   Formula for resultant magnitude: $R = \sqrt{A^2 + B^2 + 2AB \cos \theta}$ where $A$ and $B$ are magnitudes, and $\theta$ is the angle between them.

3. Vector Resolution:

   • A vector can be resolved into components along perpendicular axes:
     • Horizontal Component: $A_x = A \cos \theta$
     • Vertical Component: $A_y = A \sin \theta$

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Measurement of Physical Quantities

1. Base Quantities:

   • Measured using standard instruments.
   • Length: Ruler, Vernier Caliper.
   • Mass: Balance.
   • Time: Stopwatch, Atomic Clock.

2. Derived Quantities:

   • Calculated using formulas.
   • Example: Velocity = $\frac{\text{Displacement}}{\text{Time}}$.

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Numerical Example

1. Example: Two forces of $10 \, \text{N}$ and $20 \, \text{N}$ act at an angle of $60^\circ$. Find the resultant force.
   • Using the formula: $R = \sqrt{A^2 + B^2 + 2AB \cos \theta}$ $R = \sqrt{10^2 + 20^2 + 2(10)(20)\cos 60^\circ}$ $R = \sqrt{100 + 400 + 200 \times 0.5}$ $R = \sqrt{100 + 400 + 100} = \sqrt{600} \approx 24.49 \, \text{N}$

Vector Operations

1. Addition of Vectors:

   • Analytical Method:

     • Resolve vectors into their components along $x$ and $y$ axes.
     • Add corresponding components: $R_x = A_x + B_x, \, R_y = A_y + B_y$
     • Resultant magnitude: $R = \sqrt{R_x^2 + R_y^2}$
     • Direction of resultant: $\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

   • Special Cases:

     • Same Direction: $R = A + B$
     • Opposite Direction: $R = |A - B|$

2. Subtraction of Vectors:

   • Subtract components of one vector from the other: $\vec{R} = \vec{A} - \vec{B}$

3. Multiplication of Vectors:

   • Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \theta$

     • Produces a scalar quantity.
     • Example: Work done $W = \vec{F} \cdot \vec{d}$.

   • Cross Product: $\vec{A} \times \vec{B} = AB \sin \theta \, \hat{n}$

     • Produces a vector quantity.
     • Example: Torque $\vec{\tau} = \vec{r} \times \vec{F}$.

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Derived Quantities and Their SI Units

Derived Quantity	Formula	SI Unit	Dimensional Formula
Area	$L \times L$	$m^2$	$[L^2]$
Volume	$L \times L \times L$	$m^3$	$[L^3]$
Density	$\frac{\text{Mass}}{\text{Volume}}$	$kg \, m^{-3}$	$[M L^{-3}]$
Speed/Velocity	$\frac{\text{Distance}}{\text{Time}}$	$m \, s^{-1}$	$[L T^{-1}]$
Acceleration	$\frac{\text{Velocity}}{\text{Time}}$	$m \, s^{-2}$	$[L T^{-2}]$
Force	$Mass \times Acceleration$	$N$ (Newton)	$[M L T^{-2}]$
Pressure	$\frac{\text{Force}}{\text{Area}}$	$Pa$ (Pascal)	$[M L^{-1} T^{-2}]$
Energy/Work	$Force \times Displacement$	$J$ (Joule)	$[M L^2 T^{-2}]$

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Physical Constants

1. Definition:

   • Fundamental quantities with fixed numerical values.

2. Examples:

   • Speed of Light ($c$): $3 \times 10^8 \, m/s$.
   • Gravitational Constant ($G$): $6.67 \times 10^{-11} \, N \, m^2 \, kg^{-2}$.
   • Planck's Constant ($h$): $6.626 \times 10^{-34} \, J \, s$.

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Dimensional Consistency

1. Principle:

   • All terms in a physical equation must have the same dimensions.

2. Applications:

   • Checking Validity of Equations:
     • Example: Verify $v^2 = u^2 + 2as$.
       • Dimensions of $v^2$, $u^2$: $[L^2 T^{-2}]$.
       • Dimensions of $2as$: $[L^2 T^{-2}]$.
       • Equation is dimensionally consistent.

3. Limitations:

   • Does not confirm numerical constants.
   • Cannot verify equations involving non-dimensional terms like trigonometric functions.

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Practical Examples of Physical Quantities

1. Example 1: A car accelerates uniformly from $5 \, m/s$ to $15 \, m/s$ over a distance of $50 \, m$. Find the acceleration.

   • Formula: $v^2 = u^2 + 2as$
   • Rearranging: $a = \frac{v^2 - u^2}{2s}$ $a = \frac{15^2 - 5^2}{2 \times 50} = \frac{225 - 25}{100} = 2 \, m/s^2$

2. Example 2: Calculate the work done in moving a $10 \, kg$ object $5 \, m$ with a force of $20 \, N$ at an angle of $30^\circ$ to the horizontal.

   • Work done: $W = Fd \cos \theta$ $W = 20 \cdot 5 \cdot \cos 30^\circ = 100 \cdot \frac{\sqrt{3}}{2} = 50\sqrt{3} \, J$

Errors in Measurement of Physical Quantities

1. Definition of Error:

   • The difference between the true value and the measured value of a quantity.

2. Types of Errors:

   • Systematic Errors:
     • Arise from faulty instruments or consistent biases.
     • Example: Incorrect calibration of a scale.
   • Random Errors:
     • Caused by unpredictable fluctuations in measurement.
     • Example: Variations in stopwatch readings.
   • Gross Errors:
     • Result from human mistakes.
     • Example: Misreading an instrument.

3. Quantification of Error:

   • Absolute Error: $\Delta A = |A_{\text{measured}} - A_{\text{true}}|$
   • Relative Error: $\text{Relative Error} = \frac{\Delta A}{A_{\text{true}}}$
   • Percentage Error: $\text{Percentage Error} = \left( \frac{\Delta A}{A_{\text{true}}} \right) \times 100$

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Propagation of Errors

1. For Addition/Subtraction:

   • Total absolute error: $\Delta R = \Delta A + \Delta B$
     • Example: If $A = 5 \pm 0.2$ and $B = 3 \pm 0.1$, then: $R = A + B = 8 \pm 0.3$

2. For Multiplication/Division:

   • Total relative error: $\frac{\Delta R}{R} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$
     • Example: If $A = 5 \pm 0.2$ and $B = 3 \pm 0.1$, then: $\frac{\Delta R}{R} = \frac{0.2}{5} + \frac{0.1}{3} = 0.04 + 0.0333 \approx 0.0733$ $\Delta R = 0.0733 \cdot (A \cdot B)$

3. For Powers:

   • Error is multiplied by the exponent: $\frac{\Delta R}{R} = n \cdot \frac{\Delta A}{A}$

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Significant Figures

1. Definition:

   • Digits in a measurement that carry meaningful information about precision.

2. Rules for Significant Figures:

   • All non-zero digits are significant.
   • Zeros between significant digits are significant.
   • Trailing zeros in a decimal are significant.
   • Leading zeros are not significant.

3. Arithmetic with Significant Figures:

   • Addition/Subtraction:
     • Result has the same number of decimal places as the term with the least decimal places.
   • Multiplication/Division:
     • Result has the same number of significant figures as the term with the least significant figures.

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Practical Applications

1. Dimensional Analysis in Real Life:

   • Verifying equations.
   • Deriving new formulas.
   • Checking the consistency of units in calculations.

2. Examples of Derived Quantities:

   • Kinetic Energy: $KE = \frac{1}{2} mv^2$
     • Dimensional Formula: $[M L^2 T^{-2}]$.
   • Momentum: $p = mv$
     • Dimensional Formula: $[M L T^{-1}]$.
   • Electric Charge: $Q = I \cdot t$
     • Dimensional Formula: $[I T]$.

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Numerical Examples

1. Example 1: The mass of an object is measured as $5.0 \pm 0.1 \, kg$ and its volume as $2.0 \pm 0.2 \, m^3$. Calculate the density and its error.

   • Density: $\rho = \frac{m}{V} = \frac{5.0}{2.0} = 2.5 \, kg/m^3$
   • Relative Error: $\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + \frac{\Delta V}{V} = \frac{0.1}{5.0} + \frac{0.2}{2.0} = 0.02 + 0.1 = 0.12$
   • Absolute Error: $\Delta \rho = 0.12 \cdot 2.5 = 0.3 \, kg/m^3$
   • Final Result: $\rho = 2.5 \pm 0.3 \, kg/m^3$

2. Example 2: A sphere has a radius of $10.0 \pm 0.1 \, cm$. Calculate its volume and error.

   • Volume: $V = \frac{4}{3} \pi r^3$ $V = \frac{4}{3} \cdot 3.1416 \cdot (10)^3 = 4188.79 \, cm^3$
   • Relative Error: $\frac{\Delta V}{V} = 3 \cdot \frac{\Delta r}{r} = 3 \cdot \frac{0.1}{10} = 0.03$
   • Absolute Error: $\Delta V = 0.03 \cdot 4188.79 = 125.66 \, cm^3$
   • Final Result: $V = 4188.79 \pm 125.66 \, cm^3$

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Recap: Key Points to Remember

• Physical quantities are classified as fundamental or derived.
• Errors in measurement are unavoidable; they can be minimized but not eliminated.
• Dimensional analysis is a powerful tool to verify equations and derive relations.
• Significant figures convey the precision of a measurement.