Given below are two statements: Statement (I): The dimensions of Planck's constant and angular momentum are same. Statement (II): In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant. In the light of the above statements, choose the most appropriate answer from the options given below:

Solution & Explanation

### Related Formula E = hf implies [h] = frac[E][f] = fractextMtextL^2textT^-2textT^-1 = textMtextL^2textT^-1 L = mvr implies [L] = textM cdot (textLtextT^-1) cdot textL = textMtextL^2textT^-1 L = fracnh2pi ### Core Logic Statement I: Comparing the dimensional formula of Planck's constant (h) and angular momentum (L), both are identical [textMtextL^2textT^-1]. Hence, Statement I is correct. Statement II: According to Bohr's second postulate, angular momentum is an integral multiple of frach2pi, not an integral multiple of h. Hence, Statement II is incorrect. ### Pattern Recognition Watch out for exact definitions in standard postulates. Bohr's model requires angular momentum to be quantized in units of hbar = frach2pi, making statement II a classic trap. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 12 Physics: Atoms

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Units And Measurements

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Q44 jee_main_2024_01_february_morning Error Analysis
The radius (r), length (l) and resistance (R) of a metal wire was measured in the laboratory as: r = (0.35 pm 0.05)mathrm~cm R = (100 pm 10)mathrm~Omega l = (15 pm 0.2)mathrm~cm The percentage error in resistivity of the material of the wire is:
  • A. 25.6%
  • B. 39.9%
  • C. 37.3%
  • D. 35.6%

Solution

### Related Formula Resistivity formula: rho = RfracAl = Rfracpi r^2l Maximum relative error equation: fracDeltarhorho = fracDelta RR + 2fracDelta rr + fracDelta ll ### Core Logic Substitute the measured fractions: fracDeltarhorho = frac10100 + 2left(frac0.050.35right) + frac0.215 fracDeltarhorho = frac110 + 2left(frac17right) + frac175 fracDeltarhorho = 0.1 + 0.2857 + 0.0133 = 0.399 ### Step 1: Convert to Percentage \% severance = 0.399 times 100\% = 39.9\% ### Pattern Recognition The power of 2 in 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
Q45 jee_main_2024_01_february_morning Dimensional Analysis
The dimensional formula of angular impulse is:
  • A. [mathrmM mathrmL^-2mathrmT^-1]
  • B. [mathrmM mathrmL^2mathrmT^-2]
  • C. [mathrmM mathrmL mathrmT^-1]
  • D. [mathrmM mathrmL^2mathrmT^-1]

Solution

### Related Formula Angular Impulse equation: textAngular Impulse = int tau \, dt = Delta L textAngular 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]: [m] = [mathrmM] [v] = [mathrmLmathrmT^-1] [r] = [mathrmL] ### Step 1: Combine Dimensions [textAngular Impulse] = [mathrmM] times [mathrmLmathrmT^-1] times [mathrmL] = [mathrmMmathrmL^2mathrmT^-1] ### Pattern Recognition Angular impulse shares the exact same dimensional signature as Planck's constant (h) and angular momentum (L). ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 11 Physics: System of Particles and Rotational Motion
Q33 jee_main_2024_29_january_evening Error Analysis
A physical quantity Q is found to depend on quantities a, b, c by the relation Q = fraca^4 b^3c^2. The percentage error in a, b and c are 3\%, 4\% and 5\% respectively. Then, the percentage error in Q is:
  • A. 66\%
  • B. 43\%
  • C. 34\%
  • D. 14\%

Solution

### Related Formula For a quantity Q = fraca^x b^yc^z, the maximum relative error is: fracDelta QQ = x fracDelta aa + y fracDelta bb + z fracDelta cc Multiplying by 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) ### Core Logic Given formula: Q = fraca^4 b^3c^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) ### Step 1: Substitute the Percentage Errors Substitute the given values: * \%text error in a = 3\% * \%text error in b = 4\% * \%text error in c = 5\% \%text error in Q = 4(3\%) + 3(4\%) + 2(5\%) \%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
Q38 jee_main_2024_27_jan_morning Measuring Instruments
Identify the physical quantity that cannot be measured using a spherometer:
  • A. textRadius of curvature of concave surface
  • B. textSpecific rotation of liquids
  • C. textThickness of thin plates
  • D. textRadius of curvature of convex surface

Solution

### 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
Q47 jee_main_2024_27_jan_morning Dimensional Analysis
Given below are two statements: Statement (I): Planck's constant and angular momentum have same dimensions. Statement (II): Linear momentum and moment of force have same dimensions. In the light of the above statements, choose the correct answer from the options given below:
  • A. textStatement I is true but Statement II is false
  • B. textBoth Statement I and Statement II are false
  • C. textBoth Statement I and Statement II are true
  • D. textStatement I is false but Statement II is true

Solution

### Core Logic Evaluate dimensions step-by-step: 1. **Planck's constant (h)**: E = h u implies [h] = frac[E][ u] = fractextML^2textT^-2textT^-1 = textML^2textT^-1 2. **Angular momentum (L)**: L = mvr implies [L] = textM cdot textLT^-1 cdot textL = textML^2textT^-1 Since [h] = [L], Statement I is true. 3. **Linear momentum (P)**: P = mv implies [P] = textMLT^-1 4. **Moment of force (Torque tau)**: tau = F r implies [tau] = textMLT^-2 cdot textL = textML^2textT^-2 Since [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

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