Consider a modified Bernoulli equation. left(mathrmP+fracmathrmAmathrmBt^2right)+rhomathrmg(mathrmh+mathrmBt)+frac12rhomathrmV^2=textconstant If t has the dimension of time then the dimensions of A and B are ____, ____ respectively.

Solution & Explanation

### Related Formula textPrinciple of Homogeneity: Only quantities with same dimensions can be added/subtracted. ### Core Logic From the term (h + Bt), h (height/length) and Bt must have the same dimension. [h] = [Bt] [B] = frac[h][t] = frac[L][T] = [L T^-1] So, B = [M^0 L T^-1]. ### Step 1: Finding Dimension of A From the term left(P + fracABt^2right), pressure P and fracABt^2 must have the same dimension. Dimension of Pressure P = fractextForcetextArea = [M L^-1 T^-2]. [P] = left[fracABt^2right] [M L^-1 T^-2] = frac[A][L T^-1] times [T^2] [M L^-1 T^-2] = frac[A][L T] [A] = [M L^-1 T^-2] times [L T] = [M L^0 T^-1] So, A = [M L^0 T^-1]. ### Pattern Recognition Apply the Principle of Homogeneity across any (X + Y) terms inside brackets. Extract one unknown, plug it into the next additive bracket group. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 11 Physics: Mechanical Properties of Fluids

Reference Study Guides

Units And Measurements

More Units and Measurements Previous-Year Questions — Page 6

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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