}]]}]}]}] 30:T1cced,

### How to Use This Cheatsheet This cheatsheet is designed for rapid revision and score maximization. - **[H]**: High-frequency, must-score topics. Prioritize. - **[M]**: Medium-frequency, important for good rank. - **[L]**: Less frequent but can be tricky, don't ignore. - Read through concepts, then memorize formulas, then practice numerical patterns. - Use Assertion-Reasoning sections to train critical thinking. - Regularly review edge cases and common traps. ### Units, Dimensions & Errors #### Units & Dimensions [H] - **Fundamental Quantities (7):** Mass (kg), Length (m), Time (s), Electric Current (A), Temperature (K), Luminous Intensity (cd), Amount of Substance (mol). - **Derived Units:** Combinations of fundamental units. - **Dimensional Formula:** $[M^a L^b T^c A^d K^e ...]$. - **Principle of Homogeneity:** Dimensions of each term in an equation must be the same. Used to check correctness of equations, derive relations. - **Same Physical Quantity:** Only quantities with same dimensions can be added or subtracted. - **Dimensionless Quantities:** Angle, strain, specific gravity, refractive index, Reynolds number. - Can be unitless (strain) or have units (angle: radian). - **Trigonometric, Logarithmic, Exponential functions:** Arguments must be dimensionless. - **Common Dimensions:** - Velocity: $[LT^{-1}]$ - Acceleration: $[LT^{-2}]$ - Force: $[MLT^{-2}]$ (Newton) - Work/Energy/Torque: $[ML^2T^{-2}]$ (Joule) - Power: $[ML^2T^{-3}]$ (Watt) - Pressure/Stress/Elastic Modulus: $[ML^{-1}T^{-2}]$ (Pascal) - Momentum/Impulse: $[MLT^{-1}]$ - Surface Tension: $[MT^{-2}]$ - Planck's Constant $(h)$: $[ML^2T^{-1}]$ - Permittivity of Free Space $(\epsilon_0)$: $[M^{-1}L^{-3}T^4A^2]$ - Permeability of Free Space $(\mu_0)$: $[MLT^{-2}A^{-2}]$ - Resistance $(R)$: $[ML^2T^{-3}A^{-2}]$ - Capacitance $(C)$: $[M^{-1}L^{-2}T^4A^2]$ - Inductance $(L)$: $[ML^2T^{-2}A^{-2}]$ #### Errors [H] - **Absolute Error ($\Delta A$):** $|A_{mean} - A_i|$. - **Mean Absolute Error ($\overline{\Delta A}$):** Average of absolute errors. - **Relative Error:** $\frac{\overline{\Delta A}}{A_{mean}}$. - **Percentage Error:** $\frac{\overline{\Delta A}}{A_{mean}} \times 100\%$. - **Propagation of Errors:** - **Addition/Subtraction:** $Z = A \pm B \implies \Delta Z = \Delta A + \Delta B$. (Absolute errors add) - **Multiplication/Division:** $Z = AB$ or $Z = A/B \implies \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$. (Relative errors add) - **Power:** $Z = A^n \implies \frac{\Delta Z}{Z} = n \frac{\Delta A}{A}$. - **General:** If $Z = A^p B^q / C^r \implies \frac{\Delta Z}{Z} = p \frac{\Delta A}{A} + q \frac{\Delta B}{B} + r \frac{\Delta C}{C}$. - **Least Count (LC):** Smallest value that can be measured by an instrument. Error is often $\pm LC$ or $\pm LC/2$. - **Significant Figures:** 1. All non-zero digits are significant. 2. Zeros between non-zero digits are significant. 3. Leading zeros (before non-zero digits) are NOT significant. 4. Trailing zeros (at end of number) are significant if the number contains a decimal point. 5. Trailing zeros are NOT significant if no decimal point (e.g., 100 has 1 sig fig, 100.0 has 4). - **Addition/Subtraction:** Result has same number of decimal places as the number with the fewest decimal places. - **Multiplication/Division:** Result has same number of significant figures as the number with the fewest significant figures. - **Rounding Off:** 5 followed by non-zero digits: round up. 5 followed by zeros (or nothing): round up if preceding digit is odd, keep as is if even. - **[Insight]:** Erro