MERIT YARD
CHAPTER 1: UNITS & MEASUREMENTS
1. System of Units
A physical quantity consists of a numerical magnitude and a unit.
- Fundamental Quantities: Independent of other physical quantities (e.g., Mass, Length, Time).
- Derived Quantities: Derived from fundamental quantities (e.g., Velocity, Force).
The 7 Fundamental SI Units
| Quantity | SI Unit | Symbol |
|---|---|---|
| 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 |
🔥 EXAM POINT:
Plane Angle (Radian) and Solid Angle (Steradian) are Supplementary Units. They have Units but NO Dimensions.
2. Astronomical Units
Direct values frequently asked in exams:
- Astronomical Unit (AU): Avg. distance between Earth & Sun.
$1 \text{ AU} = 1.496 \times 10^{11} \text{ m}$
- Light Year (ly): Distance travelled by light in vacuum in 1 year.
$1 \text{ ly} = 9.46 \times 10^{15} \text{ m}$
- Parsec (pc): Largest practical unit of distance.
$1 \text{ pc} = 3.08 \times 10^{16} \text{ m} = 3.26 \text{ ly}$
3. Dimensions & Formulae
Represented as $[M^a L^b T^c]$.
| Quantity | Formula | Dimension |
|---|---|---|
| Velocity | Disp / Time | $[L T^{-1}]$ |
| Acceleration | Vel / Time | $[L T^{-2}]$ |
| Force | m × a | $[M L T^{-2}]$ |
| Work / Energy | F × d | $[M L^2 T^{-2}]$ |
| Power | W / t | $[M L^2 T^{-3}]$ |
| Pressure | F / A | $[M L^{-1} T^{-2}]$ |
| Grav. Const (G) | $F r^2 / m_1 m_2$ | $[M^{-1} L^3 T^{-2}]$ |
| Planck's (h) | $E / \nu$ | $[M L^2 T^{-1}]$ |
⚠️ IMPORTANT TRICK:
Quantities having the Same Dimensions:
- 1. $[M L^2 T^{-2}]$ $\rightarrow$ Work, Energy, Torque, Heat.
- 2. $[M L^{-1} T^{-2}]$ $\rightarrow$ Pressure, Stress, Modulus of Elasticity.
- 3. Dimensionless $\rightarrow$ Angle, Strain, Relative Density, Refractive Index.
4. Principle of Homogeneity
We can ADD or SUBTRACT only those physical quantities which have the SAME DIMENSIONS.
If $A = B + C - D$, then:
$[A] = [B] = [C] = [D]$
5. Errors in Measurement
1. Absolute Error
$\Delta a_i = |a_{mean} - a_i|$
2. Fractional Error
Error = $\frac{\Delta a_{mean}}{a_{mean}}$
3. Percentage Error
% Error = $\frac{\Delta a_{mean}}{a_{mean}} \times 100$
Combination of Errors
If $Z = \frac{A^p B^q}{C^r}$
$\frac{\Delta Z}{Z}\% = p\left(\frac{\Delta A}{A}\%\right) + q\left(\frac{\Delta B}{B}\%\right) + r\left(\frac{\Delta C}{C}\%\right)$
Note: Errors are ALWAYS ADDED.
6. Significant Figures
- Rule 1: All non-zero digits are significant.
(e.g., $245 \rightarrow 3$ sig. figs) - Rule 2: Zeros between non-zero digits are significant.
(e.g., $2005 \rightarrow 4$) - Rule 3: If number < 1, zeros to right of decimal but left of non-zero digit are NOT significant.
(e.g., $0.0052 \rightarrow 2$) - Rule 4: Terminal zeros without decimal are NOT significant.
(e.g., $4500 \rightarrow 2$) - Rule 5: Trailing zeros WITH a decimal ARE significant.
(e.g., $3.500 \rightarrow 4$) - Rule 6: Powers of 10 do not count.
(e.g., $2.5 \times 10^5 \rightarrow 2$)
7. Measuring Instruments
Vernier Caliper
L.C = 1 M.S.D - 1 V.S.D
Screw Gauge
L.C = $\frac{\text{Pitch}}{\text{Total divisions}}$
