Complete Study Material for JEE, NEET, CBSE Board Class 11 Exams
Understanding dimensional analysis is crucial in physics, especially for students preparing for exams like JEE, NEET, and CBSE Board Class 11. This concept helps in verifying equations, converting units, and checking the correctness of derived formulas. Below is a comprehensive explanation of quantities having the same dimensions, categorized systematically.
Thermal capacity, gas constant, Boltzmann constant, entropy
12
M⁰L⁰T¹
L/R, √LC, RC (L = Inductance, R = Resistance, C = Capacitance)
13
M¹L²T⁻²
I2Rt, VIt, V2/Rt,qV, LI2, q2/C, CV2, Various electrical quantities involving L=inductance, C=capacitance, q=charge, R=resistance, I=Current and V=voltage
Conceptual Questions & Answers
Q1: Why is dimensional analysis important in physics?
A: Dimensional analysis helps check the correctness of equations, derive new formulas, and convert units between different measurement systems.
Q2: Why do work and torque have the same dimensions but different physical meanings?
A: Though both have dimensions M¹L²T⁻², work is a scalar quantity representing energy transfer, whereas torque is a vector quantity causing rotational motion.
Q3: What are dimensionless quantities? Give examples.
A: Dimensionless quantities have no physical dimensions. Examples include strain, refractive index, relative density, and solid angle.
MCQs with Answers & Explanation
Q1: Which of the following pairs have the same dimensions?
(a) Work and torque (b) Force and pressure (c) Momentum and energy (d) Strain and stress
Answer: (a) Work and torque Explanation: Both have dimensions M¹L²T⁻², but torque is a rotational effect, while work is energy transfer.
Q2: What is the dimensional formula of Planck’s constant?
(a) M⁰L²T⁻² (b) M¹L²T⁻¹ (c) M¹L¹T⁻² (d) M¹L⁰T⁻²
Answer: (b) M¹L²T⁻¹ Explanation: Planck’s constant has dimensions of angular momentum, which matches M¹L²T⁻¹.
Do You Know?
Energy and torque have the same dimensions but represent different physical concepts.
Pressure and stress share dimensions because they both quantify force per unit area.
Dimensional analysis is used in checking the validity of scientific equations.
Worksheet
Find the dimensional formula of energy density.
Show that surface tension and surface energy have the same dimensions.
Verify whether the equation v=u+atv = u + at is dimensionally correct.
Convert 1 Joule into ergs using dimensional analysis.
Test Paper (Marks Distribution)
Section A: Conceptual Questions (10 Marks)
Define dimensional analysis. (2 Marks)
Why is strain a dimensionless quantity? (2 Marks)
Explain the significance of Planck’s constant. (2 Marks)
Differentiate between pressure and stress. (4 Marks)
Section B: Numerical Questions (10 Marks)
Derive the dimensional formula of impulse. (5 Marks)
Show that work and torque have the same dimensions. (5 Marks)
Section C: MCQs (10 Marks)
What is the dimension of momentum? (2 Marks)
Which quantity has the dimensional formula M¹L¹T⁻²? (2 Marks)
Work and energy share the same dimensions because: (2 Marks)
Why is refractive index dimensionless? (2 Marks)
What is the SI unit of Planck’s constant? (2 Marks)
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{ question: “Which of these quantities share the same dimensions?”, options: [“Torque and Work”, “Force and Energy”, “Power and Impulse”, “Pressure and Energy”], answer: 0, explanation: “Torque (N·m) and Work (J) both have dimensions ML²T⁻².” },
{ question: “Which of the following pairs have the same dimensions?”, options: [“Electric field and Potential”, “Momentum and Energy”, “Angular Momentum and Planck’s Constant”, “Charge and Current”], answer: 2, explanation: “Angular Momentum (kg·m²/s) and Planck’s Constant (J·s) share the same dimensions ML²T⁻¹.” },
{ question: “Which of the following pairs have the same dimensions?”, options: [“Stress and Pressure”, “Velocity and Acceleration”, “Work and Force”, “Current and Charge”], answer: 0, explanation: “Stress and Pressure both have dimensions of ML⁻¹T⁻².” },
{ question: “Which physical quantity has the same dimension as Planck’s constant?”, options: [“Torque”, “Work”, “Angular Momentum”, “Force”], answer: 2, explanation: “Planck’s constant and Angular Momentum both have ML²T⁻¹ dimensions.” },
{ question: “Which of the following pairs have the same dimensions?”, options: [“Force and Power”, “Work and Energy”, “Acceleration and Velocity”, “Current and Resistance”], answer: 1, explanation: “Work and Energy both have dimensions ML²T⁻².” },
{ question: “Which of these share the same dimensions?”, options: [“Capacitance and Inductance”, “Electric Flux and Magnetic Flux”, “Electric Field and Magnetic Field”, “Resistance and Conductance”], answer: 1, explanation: “Electric Flux and Magnetic Flux both have dimensions ML³T⁻³A⁻¹.” },
{ question: “Which of the following have the same dimensions?”, options: [“Heat and Work”, “Pressure and Force”, “Momentum and Energy”, “Power and Torque”], answer: 0, explanation: “Heat and Work both have the dimensions ML²T⁻².” },
{ question: “Which of these pairs have the same dimensions?”, options: [“Inductance and Resistance”, “Electric Field and Magnetic Field”, “Current and Voltage”, “Power and Impulse”], answer: 1, explanation: “Electric Field and Magnetic Field share the same dimensions.” },
{ question: “Which of the following has the same dimensions as Force?”, options: [“Work”, “Momentum”, “Pressure × Area”, “Torque”], answer: 2, explanation: “Pressure × Area gives Force since Pressure is Force/Area.” }
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Important Points for Quick Revision
Dimensional analysis helps in checking the correctness of equations.
Quantities with the same dimensions can have different physical meanings.
Dimensionless quantities do not have fundamental dimensions.
Conversion of units is simplified using dimensional formulas.
MCQs and conceptual questions are crucial for competitive exams.
