Ch 01 - Units And Measurements11th Physics

Measurement is the process of comparing a physical quantity with a certain basic, arbitrarily chosen, internationally accepted reference standard called a unit.

It is expressed as:
Result = Numerical Measure $\times$ Unit

Example: In 5.2 meters, '5.2' is the numerical measure and 'meters' is the unit.

• Fundamental Quantities: Basic, independent physical quantities that cannot be expressed in terms of other quantities (e.g., Length, Mass, Time).
• Derived Quantities: Physical quantities that are formed by mathematical combinations of fundamental quantities (e.g., Area = length $\times$ length, Speed = length/time).

Before the SI system, scientists used three main systems based on length, mass, and time:

• CGS System: Centimeter (cm), Gram (g), Second (s)
• FPS (British) System: Foot (ft), Pound (lb), Second (s)
• MKS System: Meter (m), Kilogram (kg), Second (s)

1. Length $\rightarrow$ Meter (m)
2. Mass $\rightarrow$ Kilogram (kg)
3. Time $\rightarrow$ Second (s)
4. Electric Current $\rightarrow$ Ampere (A)
5. Thermodynamic Temperature $\rightarrow$ Kelvin (K)
6. Amount of Substance $\rightarrow$ Mole (mol)
7. Luminous Intensity $\rightarrow$ Candela (cd)

Variables & Units:

• $\theta$ = Plane angle (rad)
• s = Arc length (m)
• r = Radius (m)

Conditions/Notes:

• The SI supplementary unit is the Radian (rad).
• It is a dimensionless quantity.
• A full circle subtends $2\pi$ radians.

• Kilogram (kg): Defined by fixing the Planck constant ($h$).
• Metre (m): Defined by fixing the speed of light in vacuum ($c$).
• Second (s): Based on the unperturbed ground-state hyperfine transition frequency of the Caesium-133 atom ($\Delta u_{Cs}$).

Variables & Units:

• \Omega = Solid angle (sr)
• A = Intercepted spherical area (m²)
• r = Radius (m)

Conditions/Notes:

• The SI supplementary unit is the Steradian (sr).
• It is a dimensionless quantity.
• A full sphere subtends $4\pi$ steradians.

• Astronomical Unit (AU): The average distance between the Earth and the Sun ($1 \text{ AU} \approx 1.5 \times 10^{11} \text{ m}$).
• Light Year (ly): The distance that light travels in a vacuum in one year ($1 \text{ ly} \approx 9.46 \times 10^{15} \text{ m}$).
• Parsec (pc): The distance at which an arc of 1 AU subtends a parallax angle of 1 arc-second ($1 \text{ pc} \approx 3.08 \times 10^{16} \text{ m}$ or $3.26 \text{ ly}$).

• 1 Fermi (fm): $10^{-15} \text{ m}$. Used for measuring nuclear sizes.
• 1 Angstrom (\AA): $10^{-10} \text{ m}$. Used for measuring atomic sizes and optical wavelengths.
• 1 Barn (b): $10^{-28} \text{ m}^2$. Used for measuring extremely small nuclear cross-sectional areas.

Express the number in Scientific Notation: $a \times 10^b$ (where $1 \le a \le 10$).

Rules for rounding 'a':

• If $a \le 5$, round down. The order of magnitude is $10^b$.
• If $a > 5$, round up. The order of magnitude is $10^{b+1}$.

Example: Earth's diameter is $1.28 \times 10^7$ m. Since $1.28 \le 5$, the order of magnitude is $10^7$.

• Accuracy: The degree of closeness of a measured value to the true or standard value (e.g., hitting the bullseye).
• Precision: The degree of closeness of multiple repeated measurements to each other under identical conditions, irrespective of the true value. It often depends on the instrument's least count.

Note: A measurement can be highly precise without being accurate.

• Systematic Errors: Consistent, predictable errors that occur in the same direction (always positive or always negative). Often caused by instrumental flaws (e.g., zero error) or flawed technique.
• Random Errors: Unpredictable, irregular fluctuations in measurements due to environmental changes or human limitations.
• Gross Errors: Errors caused by sheer human carelessness, misreading, or improper setup.

• Absolute Error ($\Delta a$): The magnitude of the difference between the individual measured value and the true (mean) value of the quantity.
• Relative (Fractional) Error: The ratio of the mean absolute error to the true (mean) value of the quantity measured.
• Percentage Error: Relative error expressed as a percentage (Relative Error $\times$ 100%).

• Addition & Subtraction: The Absolute Errors are added together.
  • Rule: If $Z = A + B$ or $Z = A - B$, then $\Delta Z = \Delta A + \Delta B$
• Multiplication & Division: The Relative (or Fractional) Errors are added together.
  • Rule: If $Z = AB$ or $Z = A/B$, then $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$

Variables & Units:

• v = Velocity (m/s)
• a = Acceleration (m/s²)

Conditions/Notes:

• Mass [M] has a zero exponent in both quantities.
• Speed and Velocity share the exact same dimensional formula.

Significant figures are all the digits in a measurement that are known reliably (certain) plus the first digit that is uncertain.

Example: In the measurement 1.62 s, the digits 1 and 6 are certain, while the 2 is uncertain. The total number of significant figures is three.

Variables & Units:

• F = Force (Newton, N or kg·m/s²)

Conditions/Notes:

• This is the most foundational dimensional formula in mechanics. Memorize it to easily derive others (like Work and Pressure).

Exact numbers, multiplying factors, or dividing factors that are not measured values possess an infinite number of significant figures.

Note: Because they have infinite precision, they do not limit or dictate the number of significant figures in the final calculated result.

Variables & Units:

• W or E = Work / Energy (Joule, J or kg·m²/s²)

Conditions/Notes:

• All forms of energy (kinetic, potential, elastic) share this exact dimension.
• Torque also shares this dimension, despite being a different physical vector quantity.

• Between Non-Zeros: Always significant (e.g., 1002 has 4 sig figs).
• Leading Zeros: NEVER significant; they only locate the decimal point (e.g., 0.0023 has 2 sig figs).
• Trailing Zeros WITHOUT a Decimal: NOT significant (e.g., 1200 has 2 sig figs).
• Trailing Zeros WITH a Decimal: ALWAYS significant (e.g., 12.00 has 4 sig figs).

Variables & Units:

• P = Power (Watt, W or kg·m²/s³)

Conditions/Notes:

• Represents the rate of doing work or transferring energy.

When the insignificant digit to be dropped is exactly 5, look at the preceding digit:

• If it is Even, leave it unchanged (e.g., $2.745 \rightarrow 2.74$).
• If it is Odd, round it up by 1 (e.g., $2.735 \rightarrow 2.74$).

Note: If the digit is $>5$, always round up. If $<5$, always round down.

Variables & Units:

• P = Pressure / Stress (Pascal, Pa or N/m²)

Conditions/Notes:

• Pressure and Stress have identical dimensions because both are defined as Force per unit Area.

Multiplication & Division:
The final result should retain as many significant figures as there are in the original number with the fewest significant figures.
Example: $4.237 \div 2.51 = 1.69$ (Rounded to 3 sig figs because 2.51 has 3).

Addition & Subtraction:
The final result should retain as many decimal places as there are in the original number with the fewest decimal places (NOT total significant figures).
Example: $436.32 + 227.2 + 0.301 = 663.8$ (Rounded to 1 decimal place because 227.2 has 1).

The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that specific quantity.

• They are denoted by square brackets [ ].
• In mechanics, all quantities can be written in terms of [L] (Length), [M] (Mass), and [T] (Time).
• Example: Volume is $[L^3]$. It possesses zero dimension in mass $[M^0]$ and zero in time $[T^0]$.

Dimensionless quantities have zero power for all base dimensions, mathematically represented as $[M^0 L^0 T^0]$.

Common Examples:

• Mathematical: Pure numbers, $\pi$, trigonometric ratios ($\sin \theta$, $\cos \theta$), exponential, and logarithmic functions.
• Physical Ratios: Plane angle (radian), solid angle (steradian), refractive index, relative density, and Reynolds number.

Note: A quantity can have units (like radians) but still be completely dimensionless.

Variables & Units:

• G = Gravitational Constant (N·m²/kg²)

Conditions/Notes:

• Very common test question. Notice the negative mass exponent.
• Even though it's a constant, it has units and dimensions, unlike pure numbers.

Variables & Units:

• $\Delta Z / Z$ = Maximum relative error in Z
• p, q, r = Exponents of the respective quantities

Conditions/Notes:

• The relative error of a quantity raised to a power is the power multiplied by the relative error of the individual quantity.
• Crucial Rule: Errors ALWAYS ADD to find the maximum possible error. Even though $C$ is in the denominator, its error contribution is added, never subtracted.

Statement: Physical quantities can only be added, subtracted, or equated if they have the same dimensions.

If an equation is written as $A = B + C$, then the dimensions of $A$, $B$, and $C$ must all be completely identical.

Significance: This is the core principle used to verify the dimensional consistency of formulas.

By writing out the dimensions of every single term on the LHS and RHS of an equation. If all terms match, the equation is dimensionally consistent.

Example: Kinematic equation $v = v_0 + at$

• $[v] = [L T^{-1}]$
• $[v_0] = [L T^{-1}]$
• $[at] = [L T^{-2}] \times [T] = [L T^{-1}]$

Because every separated term is $[L T^{-1}]$, the equation passes the consistency test.

Warning: Dimensional consistency does not guarantee the equation is physically correct (it ignores dimensionless constants).

1. Identify the variables the physical quantity depends on.
2. Assume a product-type dependence with unknown exponents (e.g., $T = k \cdot l^x g^y m^z$).
3. Apply the Principle of Homogeneity by equating the dimensions on the LHS and RHS.
4. Solve the resulting linear equations for the unknown exponents ($x, y, z$).

Note: The dimensionless constant $k$ must be found experimentally.

Variables & Units:

• p = Momentum (kg·m/s)
• J = Impulse (N·s)

Conditions/Notes:

• Momentum and Impulse share the exact same dimensional formula because Impulse is defined as the change in momentum (Impulse-Momentum Theorem).

• It cannot determine the value of dimensionless constants (like $2\pi$, $\frac{1}{2}$, or $e$).
• It fails completely for equations involving trigonometric, logarithmic, or exponential functions (since their arguments must be dimensionless).
• It cannot differentiate between distinct physical quantities that share the same dimensions (e.g., Work and Torque).
• It is limited to deducing relationships that are products of powers (it cannot deduce relations with addition/subtraction like $v^2 = u^2 + 2as$ directly).