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.
Physical Quantities with Same Dimensions
| S. No. | Dimension | Quantities |
|---|---|---|
| 1 | M⁰L⁰T⁻¹ | Frequency, angular frequency, angular velocity, velocity gradient, decay constant |
| 2 | M¹L²T⁻² | Work, internal energy, potential energy, kinetic energy, torque, moment of force |
| 3 | M¹L⁻¹T⁻² | Pressure, stress, Young’s modulus, bulk modulus, modulus of rigidity, energy density |
| 4 | M¹L¹T⁻¹ | Momentum, impulse |
| 5 | M⁰L¹T⁻² | Acceleration due to gravity, gravitational field intensity |
| 6 | M¹L¹T⁻² | Thrust, force, weight, energy gradient |
| 7 | M¹L²T⁻¹ | Angular momentum, Planck’s constant |
| 8 | M¹L⁰T⁻² | Surface tension, surface energy (energy per unit area) |
| 9 | M⁰L⁰T⁰ | Strain, refractive index, relative density, angle, solid angle, distance gradient, relative permittivity, relative permeability |
| 10 | M⁰L²T⁻² | Latent heat, gravitational potential |
| 11 | M⁰L²T⁻²θ⁻¹ | 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)
Test Your Knowledge (Quiz)
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.
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Written by: Neeraj Anand
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