Arjuna JEE 2027: Physics - Lecture 09: Units, Measurements, and Errors

Homework Discussion: Units and Dimensions

• Question 1: Conversion of Gravitational Constant ($G$)     * Problem: Find the value of the gravitational constant in the CGS system given its SI value is $6.6 \times 10^{-11} \, kg^{-1}m^3s^{-2}$.     * Dimensional Formula: $[G] = [M^{-1}L^3T^{-2}]$.     * Conversion Formula: $n_1 u_1 = n_2 u_2 \Rightarrow n_2 = n_1 \left[ \frac{M_1}{M_2} \right]^{a} \left[ \frac{L_1}{L_2} \right]^{b} \left[ \frac{T_1}{T_2} \right]^{c}$.     * Variables:         * $n_1 = 6.6 \times 10^{-11}$.         * $M_1 = 1 \, kg$, $M_2 = 1 \, gm$.         * $L_1 = 1 \, m$, $L_2 = 1 \, cm$.         * $T_1 = 1 \, s$, $T_2 = 1 \, s$.     * Calculation:         * $n_2 = 6.6 \times 10^{-11} \left[ \frac{1000 \, gm}{1 \, gm} \right]^{-1} \left[ \frac{100 \, cm}{1 \, cm} \right]^{3} \left[ \frac{1 \, s}{1 \, s} \right]^{-2}$.         * $n_2 = 6.6 \times 10^{-11} \times 10^{-3} \times 10^6$.         * $n_2 = 6.6 \times 10^{-8}$.     * Result: The value in CGS is $6.6 \times 10^{-8} \, gm^{-1}cm^3s^{-2}$.

• Question 2: Modified Unit System for Planck's Constant ($h$)     * Problem: The value of Planck's constant is $6.6 \times 10^{-34} \, kg \, m^2s^{-1}$. Find its value in a system where length is doubled ($L_2 = 2L_1$) and time is halved ($T_2 = \frac{T_1}{2}$).     * Dimensional Formula: Based on $E = h \nu$, $[h] = [M^1 L^2 T^{-1}]$.     * Calculation:         * $n_2 = 6.6 \times 10^{-34} \left[ \frac{M_1}{M_1} \right]^{1} \left[ \frac{L_1}{2L_1} \right]^{2} \left[ \frac{T_1}{T_1 / 2} \right]^{-1}$.         * $n_2 = 6.6 \times 10^{-34} \times 1 \times \frac{1}{4} \times (2)^{-1}$.         * $n_2 = 6.6 \times 10^{-34} \times \frac{1}{8}$.         * $n_2 = 0.825 \times 10^{-34} = 8.25 \times 10^{-35}$.

Advanced Concepts: Permittivity and Permeability

• Permittivity of Free Space ($\epsilon_0$):     * Derived from Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{q^2}{r^2}$.     * Dimensions: $[\epsilon_0] = [M^{-1}L^{-3}T^4A^2]$.
• Permeability of Free Space ($\mu_0$):     * Related to magnetic fields and masses ($F = G \frac{m^2}{r^2}$ analogy used for explanation).
• Universal Relation through Speed of Light:     * The speed of light ($c$) is defined as: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.     * Dimensional check: $\frac{1}{\mu_0 \epsilon_0} = (speed)^2 = [L^2 T^{-2}]$.
• Instruction: Students are advised not to memorize extra dimensions beyond these; focusing on core formulas is more effective.

Introduction to Errors and Their Types

• Core Concepts: Errors are the uncertainties in measurements encountered during experiments.
• Systematic Errors (Known / Predictable):     * Nature: Causes are known; they tend to be in one direction (either positive or negative).     * Types:         * Instrumental Errors: Caused by faulty or incorrectly calibrated instruments (e.g., a scale always showing an extra $+0.5 \, g$).         * Environmental Errors: Caused by external conditions like temperature, pressure, or humidity (e.g., expansion of a ruler in high heat).     * Fix: Proper calibration and adjusting for known parameters.
• Random Errors (Unknown / Unpredictable):     * Nature: Causes are unknown; result in small, unpredictable fluctuations.     * Fix: Perform more trials. Take multiple readings ($n$) and find the average value.     * Relationship: Random Error is inversely proportional to the number of readings ($n$). If readings increase by a factor of $k$, the random error reduces to $1/k$ of the initial error.
• Gross Errors (Human Errors):     * Nature: Caused by human carelessness (the "Reason is you" category).     * Examples: Recording 25 instead of 52, or errors while taking observations and performing calculations.     * Fix: Be careful and attentive (motto: "Savdhani hati durghatna ghati").

Questions and Discussion on Measuring Errors

• Interaction:     * Q: When we perform experiments do we make mistakes? A: Yes.     * Q: Can there be a faulty instrument which appears to be accurate? A: Yes.     * Q: Can we make mistakes while taking reading? A: Yes.     * Q: Can there be any kind of error which is unknown to us? A: Yes.
• Question 3: Scaling Random Error     * Problem: If the random error in an experiment of 10 readings is $x$, how many readings must be taken to reduce the random error to $x/5$?     * Solution: Since Random Error $\propto \frac{1}{\text{No. of Readings}}$, to reach $x/5$, the number of readings must be 5 times the initial count. Result: $10 \times 5 = 50$ readings.

Calculation and Representation of Errors

• Experiment Example: Time Period of a Pendulum     * Readings ($S.No$): 1) $1.8 \, s$, 2) $1.9 \, s$, 3) $2.0 \, s$, 4) $2.1 \, s$, 5) $2.2 \, s$.
• Step 1: Mean Value (Average Value):     * $\bar{T} = \frac{T_1 + T_2 + T_3 + T_4 + T_5}{5}$.     * Calculation: $\frac{1.8+1.9+2.0+2.1+2.2}{5} = 2.0 \, s$.
• Step 2: Absolute Error ($\Delta T$):     * Calculated for each reading as $\text{Reading} - \text{Average Value}$.     * $\Delta T_1 = 1.8 - 2.0 = -0.2$.     * $\Delta T_2 = 1.9 - 2.0 = -0.1$.     * $\Delta T_3 = 2.0 - 2.0 = 0$.     * $\Delta T_4 = 2.1 - 2.0 = +0.1$.     * $\Delta T_5 = 2.2 - 2.0 = +0.2$.
• Step 3: Mean Absolute Error ($\overline{\Delta T}$):     * The average of the magnitudes (absolute values) of the errors.     * $\overline{\Delta T} = \frac{|\Delta T_1| + |\Delta T_2| + ... + |\Delta T_n|}{n}$.     * Calculation: $\frac{0.2+0.1+0+0.1+0.2}{5} = \frac{0.6}{5} = 0.12 \, s$.
• Step 4: Reporting the Result:     * Format: $Value = \text{Mean Value} \pm \text{Mean Absolute Error}$.     * Report: $T = 2 \pm 0.12 \, s$.
• Step 5: Fractional Error:     * $\text{Fractional Error} = \frac{\text{Mean Absolute Error}}{\text{Mean Value}}$.     * Calculation: $\frac{0.12}{2} = 0.06$.
• Step 6: Percentage Error:     * $\text{Percentage Error} = \text{Fractional Error} \times 100$.     * Calculation: $0.06 \times 100 = 6\%$.     * Note: Errors up to $5\text{-}7\%$ are generally considered acceptable.

Practical Numerical Applications

• Question 4: Reporting Length     * Given: Results reported as $5 \pm 0.015$.     * Analysis:         * Mean Value ($L_{avg}$): 5.         * Mean Absolute Error ($\Delta L$): 0.015.         * Fractional Error: $\frac{0.015}{5} = 0.003$.         * Percentage Error: $0.3\%$.
• Question 5: Refractive Index of Glass     * Data: 1.45, 1.56, 1.54, 1.44, 1.54, 1.53.     * Calculation:         * Mean ($\mu_{avg}$): $\frac{1.45+1.56+1.54+1.44+1.54+1.53}{6} = 1.51$.         * Absolute Errors: $-0.06$, $+0.05$, $+0.03$, $-0.07$, $+0.03$, $+0.02$.         * Mean Absolute Error ($\overline{\Delta \mu}$): $\frac{0.06+0.05+0.03+0.07+0.03+0.02}{6} = 0.043$.         * Report: $1.51 \pm 0.043$.         * Percentage Error: $\frac{0.043}{1.51} \times 100 \approx 2.8\%$.
• Question 6: Pendulum Oscillations and Minimum Division     * Data: 100 oscillations recorded as 90s, 91s, 95s, and 92s.     * Note: The minimum division (resolution) of the clock is 1s, which dictates the precision of the reported mean.

Homework and Resources

• Module Assignments:     * Module 1 Examples: 1 to 5.     * Check your understanding: 1 to 4.     * Topic-wise questions: Extensive list including 1-8, 12-18, 21-26, 28-36, etc.     * Advanced and PYQ levels involve 50+ specific problem numbers.
• Schedule: Full homework on Errors to be completed by next week (100+ questions).
• Promotional Note: Arjuna For JEE Main & Advanced Class 11th PCM Combo set includes 16 books with 100-150 high probability questions per chapter.