Units and Errors

Physical Quantities and SI Units

The International System of Units (SI) classifies physical quantities into Base Quantities, Derived Quantities, and Supplementary Units to provide a universal standard for measurement.

$$F = ma$$

Force is derived from Newton's Second Law, showing how derived units combine base units.

$F$=Force(newtons (N))

$m$=Mass(kilograms (kg))

$a$=Acceleration(metres per second squared (m/s²))

deriving Pressure units

→

$[P] = [F]/[A] = [MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}] = \text{Pa}$

deriving Energy units

→

$[E] = [F][d] = [MLT^{-2}][L] = [ML^2T^{-2}] = \text{J}$

Base Quantities: Seven fundamental building blocks: length (m), mass (kg), time (s), current (A), temperature (K), intensity (cd), and amount (mol).

Derived Quantities: Combinations of base units, such as Force ($1 \text{ N} = 1 \text{ kg m s}^{-2}$) or Pressure ($1 \text{ Pa} = 1 \text{ kg m}^{-1} \text{ s}^{-2}$).

Scientific Notation: Standardized format using powers of ten with exactly one non-zero digit to the left of the decimal.

The Seven SI Base Units

•

Length → metre (m)

•

Mass → kilogram (kg)

•

Time → second (s)

•

Electric Current → ampere (A)

•

Temperature → kelvin (K)

•

Luminous Intensity → candela (cd)

•

Amount of Substance → mole (mol)

Common SI Prefixes

From largest to smallest

•

giga (G) = 10⁹

•

mega (M) = 10⁶

•

kilo (k) = 10³

•

centi (c) = 10⁻²

•

milli (m) = 10⁻³

•

micro (µ) = 10⁻⁶

•

nano (n) = 10⁻⁹

•

pico (p) = 10⁻¹²

Supplementary units describe purely geometric quantities: the Radian for plane angles and the Steradian for solid angles.

$$\theta = \frac{s}{r}$$

The radian is defined by the ratio of arc length to radius.

$\theta$=Plane angle(radians (rad))

$s$=Arc length(meters (m))

$r$=Radius(meters (m))

$s = r$

→

$\theta = 1 \text{ rad} \approx 57.3^\circ$

$s = 2\pi r$

→

Full circle is $2\pi$ radians

Steradian (sr): Subtended at center of sphere by area $A = r^2$. Total solid angle of a sphere is $4\pi$ sr.

Precision, Accuracy, and Errors

Experimental results differ from true values due to Random Error, caused by unknown fluctuations, and Systematic Error, caused by faulty calibration or technique.

Accuracy: How close a measurement is to the true value; limited by systematic error.

Precision: How consistent repeated measurements are; limited by Least Count and random error.

Parallax Error: Always read measurements with eye perpendicular to scale to avoid systematic offset.

Quantifying the reliability of data through absolute, fractional, and percentage uncertainties.

$$\text{Percentage Uncertainty} = \frac{\Delta x}{x} \times 100\%$$

Relates the instrument's limitation to the magnitude of the measured quantity.

$\Delta x$=Absolute uncertainty (usually least count)(same as $x$)

$x$=Measured value(variable)

$x$ is very small

→

Percentage uncertainty becomes very high for a fixed instrument.

using $x^n$

→

$\%\text{ uncertainty in } x^n = |n| \times \%\text{ uncertainty in } x$

Products/Quotients: ADD percentage uncertainties. $\%\Delta(AB) = \%\Delta A + \%\Delta B$.

Sums/Differences: ADD absolute uncertainties. $\Delta(A \pm B) = \Delta A + \Delta B$.

Powers: Multiply percentage uncertainty by the power. $\%\Delta(x^n) = |n| \times \%\Delta x$.

Precision ≠ Accuracy: A vernier caliper (least count 0.01 cm) is more precise than a metre rule (0.1 cm), but measuring a small object like 0.45 cm gives 2% uncertainty — less accurate than 25.5 cm with a metre rule (0.4% uncertainty).

Rules of Significant Figures

Significant Figures include all accurately known digits plus the first Doubtful Digit, indicating the precision of a measurement.

Leading Zeros: Never significant (e.g., 0.002 has 1 sig fig).

Trapped Zeros: Always significant (e.g., 2.002 has 4 sig figs).

Trailing Zeros: Significant only if a decimal is present (e.g., 2.30 has 3, 230 has 2).

Scientific Notation Clarity: In $8.70 \times 10^4$ kg, the digits 8, 7, 0 are all significant (3 sig figs). Powers of ten are NOT counted as significant figures.

Ambiguous Integers: 8000 kg could have 1, 2, 3, or 4 sig figs depending on instrument precision. Writing $8.00 \times 10^3$ kg makes it unambiguous (3 sig figs).

Maintaining precision consistency during calculations requires specific rounding rules for addition/subtraction and multiplication/division.

Multiplication/Division: Result keeps the same number of sig figs as the Least Accurate Factor.

Addition/Subtraction: Result keeps the same number of Decimal Places as the least precise term.

When the digit to be dropped is exactly 5, follow the even-odd convention to avoid systematic bias in large datasets.

$$\text{If digit to drop is 5, round toward even}$$

The 'round half to even' rule prevents statistical bias when rounding many numbers.

$\text{even}$=Last retained digit is even (0, 2, 4, 6, 8)(N/A)

$\text{odd}$=Last retained digit is odd (1, 3, 5, 7, 9)(N/A)

2.345 to 3 sig figs

→

Last retained digit is 4 (even), so round to 2.34

2.355 to 3 sig figs

→

Last retained digit is 5 (odd), so round to 2.36

4.650 to 2 sig figs

→

Last retained digit is 6 (even), so round to 4.6

4.750 to 2 sig figs

→

Last retained digit is 7 (odd), so round to 4.8

Why This Rule: If we always rounded 5 up, averaging many rounded numbers would show upward bias. Even-odd balances this.