A physical quantity Q$Q$ is related to four observables a, b, c, d$a, b, c, d$ as follows: Q = fracab^4cd$Q = \frac{ab^4}{cd}$ where, a = (60 pm 3)mathrm~Pa$a = (60 \pm 3)\mathrm{~Pa}$ ; b = (20 pm 0.1)mathrm~m$b = (20 \pm 0.1)\mathrm{~m}$ ; c = (40 pm 0.2)mathrm~Nsm^-2$c = (40 \pm 0.2)\mathrm{~Nsm}^{-2}$ and d = (50 pm 0.1)mathrm~m$d = (50 \pm 0.1)\mathrm{~m}$ , then the percentage error in Q$Q$ is fracx1000$\frac{x}{1000}$ , where x = $x = $ ______.

### Related Formula Resistivity formula: rho = RfracAl = Rfracpi r^2l$$\rho = R\frac{A}{l} = R\frac{\pi r^2}{l}$$ Maximum relative error equation: fracDeltarhorho = fracDelta RR + 2fracDelta rr + fracDelta ll$$\frac{\Delta\rho}{\rho} = \frac{\Delta R}{R} + 2\frac{\Delta r}{r} + \frac{\Delta l}{l}$$ ### Core Logic Substitute the measured fractions: fracDeltarhorho = frac10100 + 2left(frac0.050.35right) + frac0.215$$\frac{\Delta\rho}{\rho} = \frac{10}{100} + 2\left(\frac{0.05}{0.35}\right) + \frac{0.2}{15}$$ fracDeltarhorho = frac110 + 2left(frac17right) + frac175$$\frac{\Delta\rho}{\rho} = \frac{1}{10} + 2\left(\frac{1}{7}\right) + \frac{1}{75}$$ fracDeltarhorho = 0.1 + 0.2857 + 0.0133 = 0.399$$\frac{\Delta\rho}{\rho} = 0.1 + 0.2857 + 0.0133 = 0.399$$ ### Step 1: Convert to Percentage \% severance = 0.399 times 100\% = 39.9\%$$\% severance = 0.399 \times 100\% = 39.9\%$$ ### Pattern Recognition The power of 2 in r^2$r^2$ doubles its fractional error impact, making it the most dominant source of error in this setup. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 12 Physics: Current Electricity

### Related Formula Angular Impulse equation: textAngular Impulse = int tau \, dt = Delta L$$\text{Angular Impulse} = \int \tau \, dt = \Delta L$$ textAngular Momentum (L) = mvr$$\text{Angular Momentum } (L) = mvr$$ ### Core Logic By the impulse-momentum theorem for rotation, angular impulse equals the net change in angular momentum. Extract dimensions from [mvr]$[mvr]$: [m] = [mathrmM]$$[m] = [\mathrm{M}]$$ [v] = [mathrmLmathrmT^-1]$$[v] = [\mathrm{L}\mathrm{T}^{-1}]$$ [r] = [mathrmL]$$[r] = [\mathrm{L}]$$ ### Step 1: Combine Dimensions [textAngular Impulse] = [mathrmM] times [mathrmLmathrmT^-1] times [mathrmL] = [mathrmMmathrmL^2mathrmT^-1]$$[\text{Angular Impulse}] = [\mathrm{M}] \times [\mathrm{L}\mathrm{T}^{-1}] \times [\mathrm{L}] = [\mathrm{M}\mathrm{L}^2\mathrm{T}^{-1}]$$ ### Pattern Recognition Angular impulse shares the exact same dimensional signature as Planck's constant (h$h$) and angular momentum (L$L$). ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 11 Physics: System of Particles and Rotational Motion

### Related Formula For a quantity Q = fraca^x b^yc^z$Q = \frac{a^x b^y}{c^z}$, the maximum relative error is: fracDelta QQ = x fracDelta aa + y fracDelta bb + z fracDelta cc$$\frac{\Delta Q}{Q} = x \frac{\Delta a}{a} + y \frac{\Delta b}{b} + z \frac{\Delta c}{c}$$ Multiplying by 100$100$ gives the percentage error equation: \%text error in Q = x(\%text error in a) + y(\%text error in b) + z(\%text error in c)$$\%\text{ error in } Q = x(\%\text{ error in } a) + y(\%\text{ error in } b) + z(\%\text{ error in } c)$$ ### Core Logic Given formula: Q = fraca^4 b^3c^2$$Q = \frac{a^4 b^3}{c^2}$$ Applying the error propagation rule: fracDelta QQ times 100 = 4 left(fracDelta aa times 100right) + 3 left(fracDelta bb times 100right) + 2 left(fracDelta cc times 100right)$$\frac{\Delta Q}{Q} \times 100 = 4 \left(\frac{\Delta a}{a} \times 100\right) + 3 \left(\frac{\Delta b}{b} \times 100\right) + 2 \left(\frac{\Delta c}{c} \times 100\right)$$ ### Step 1: Substitute the Percentage Errors Substitute the given values: * \%text error in a = 3\%$\%\text{ error in } a = 3\%$ * \%text error in b = 4\%$\%\text{ error in } b = 4\%$ * \%text error in c = 5\%$\%\text{ error in } c = 5\%$ \%text error in Q = 4(3\%) + 3(4\%) + 2(5\%)$$\%\text{ error in } Q = 4(3\%) + 3(4\%) + 2(5\%)$$ \%text error in Q = 12\% + 12\% + 10\% = 34\%$$\%\text{ error in } Q = 12\% + 12\% + 10\% = 34\%$$ ### Pattern Recognition Error percentages always add up, weighted by their exponents in the mathematical expression, regardless of whether the variable is in the numerator or denominator. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements

### Core Logic A spherometer is a mechanical instrument designed to measure small vertical displacements to calculate the thickness of thin plates or the radius of curvature of spherical (convex/concave) surfaces. Specific rotation of liquids is an optical property measured via a polarimeter, completely outside the scope of a spherometer. ### Pattern Recognition Spherometers operate purely on linear micrometer screw scale geometry, hence restricted strictly to spatial dimensions. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements

### Core Logic Evaluate dimensions step-by-step: 1. **Planck's constant (h$h$)**: E = h u implies [h] = frac[E][ u] = fractextML^2textT^-2textT^-1 = textML^2textT^-1$$E = h u \implies [h] = \frac{[E]}{[ u]} = \frac{\text{ML}^2\text{T}^{-2}}{\text{T}^{-1}} = \text{ML}^2\text{T}^{-1}$$ 2. **Angular momentum (L$L$)**: L = mvr implies [L] = textM cdot textLT^-1 cdot textL = textML^2textT^-1$$L = mvr \implies [L] = \text{M} \cdot \text{LT}^{-1} \cdot \text{L} = \text{ML}^2\text{T}^{-1}$$ Since [h] = [L]$[h] = [L]$, Statement I is true. 3. **Linear momentum (P$P$)**: P = mv implies [P] = textMLT^-1$$P = mv \implies [P] = \text{MLT}^{-1}$$ 4. **Moment of force (Torque tau$\tau$)**: tau = F r implies [tau] = textMLT^-2 cdot textL = textML^2textT^-2$$\tau = F r \implies [\tau] = \text{MLT}^{-2} \cdot \text{L} = \text{ML}^2\text{T}^{-2}$$ Since [P] neq [tau]$[P] \neq [\tau]$, Statement II is false. ### Pattern Recognition Planck's constant can always be paired with angular momentum units (Joule-seconds), while moment of force matches work/energy footprints, not translational momentum. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements