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When a derived quantity is expressed in terms of fundamental quantities, it is written as a product of different powers of the fundamental quantities. The powers to which fundamental quantities must be raised in order to express the given physical quantity are called its dimensions.
Dimensions of Fundamental Quantities
The dimensions of a physical quantity are the powers to which the unit of fundamental quantities are raised to represent that quantity.
The seven fundamental quantities and their respective dimensions are:
| S.No. | Physical Quantity | Dimension |
|---|---|---|
| 1 | Length | [L] |
| 2 | Mass | [M] |
| 3 | Time | [T] |
| 4 | Temperature | [K] |
| 5 | Electric Current | [A] |
| 6 | Luminous Intensity | [Cd] |
| 7 | Amount of Substance | [Mol] |
** Note: Two supplementary fundamental quantities that is plane angle and solid angle have no dimensions.
Understanding Dimensions with Example
To make the concept clearer, consider the physical quantity force:
Force = Mass × Acceleration
Since Velocity is given by:
Velocity = Length/Time
and acceleration is given by:
Acceleration=Velocity/Time=Length/Time2
Force = Mass x Acceleration
Therefore, Force=Mass×Length/Time2
Force=Mass×Length×(Time)−2
Thus, the dimensions of force are:
Force=[MLT−2]
Dimensional Equations and Dimensional Formula
The physical quantity that is expressed in terms of base quantities is enclosed in square brackets to indicate that the equation represents dimensions and not magnitudes.
Dimensional Equation : The equation obtained by equating a physical quantity with its dimensions formula is called dimensional equation of the given physical quantity
For Example : The dimensional equation of Force is :
[Force]=[MLT−2]
Dimensional Formula : If we consider only the right-hand side (RHS) of the equation, it is called the dimensional formula:
Dimensional Formula of Force=[MLT−2]
Important Dimensions of Physics Quantities
Here are some commonly used physical quantities with their dimensional formulas:
| Physical Quantity | Dimensional Formula |
|---|---|
| Velocity | [LT−1] |
| Acceleration | [LT−2] |
| Momentum | Mas x Velocity = [MLT−1] |
| Work or Energy | Force x Distance = [ML2T−2] |
| Power | Work/Time = [ML2T−3] |
| Pressure | Force/Area = [ML-1T−2] |
Important Questions with Answers
What Are Dimensional Constants?
The physical quantities with dimensions and a fixed value are called dimensional constants. For example, gravitational constant (G), Planck’s constant (h), universal gas constant (R), velocity of light in a vacuum (C), etc.
What Are Dimensionless Quantities?
Dimensionless quantities are those which do not have dimensions but have a fixed value.
- Dimensionless quantities without units: Pure numbers, π, e, sin θ, cos θ, tan θ etc.
- Dimensionless quantities with units: Angular displacement – radian, Joule’s constant – joule/calorie, etc.
What Are Dimensional Variables?
Dimensional variables are those physical quantities which have dimensions and do not have a fixed value. For example, velocity, acceleration, force, work, power, etc.
What Are the Dimensionless Variables?
Dimensionless variables are those physical quantities which do not have dimensions and do not have a fixed value. For example, specific gravity, refractive index, the coefficient of friction, Poisson’s ratio, etc.
FAQs (Frequently Asked Questions)
Q1: Why are dimensions important in physics?
A: Dimensions help in verifying equations, deriving new relationships, and converting units.
Q2: What is the difference between the dimensional formula and the dimensional equation?
A: The dimensional formula consists only of the dimensions of a quantity, whereas the dimensional equation shows how the quantity is derived in terms of base quantities.
Q3: Can two different physical quantities have the same dimensions?
A: Yes, for example, work and energy both have the same dimensions [ML²T⁻²].
Multiple Choice Questions (MCQs)
Q1: What is the dimensional formula of force?
(a) [MLT⁻²]
(b) [ML²T⁻²]
(c) [MLT⁻¹]
(d) [M²L²T⁻²]
Answer: (a) [MLT⁻²]
Explanation: Force is given by mass × acceleration, and acceleration is length/time².
Q2: Which of the following quantities has the dimensional formula [ML²T⁻²]?
(a) Work
(b) Power
(c) Pressure
(d) Velocity
Answer: (a) Work
Explanation: Work is given by Force × Displacement which results in the formula [ML²T⁻²].
Conceptual Questions with Answers
Q1: Can a physical equation be correct even if its dimensions do not match on both sides?
A: No, for an equation to be dimensionally correct, both sides must have the same dimensions.
Q2: Why do we use dimensions in checking the correctness of an equation?
A: Because dimensions remain unchanged in all systems of units, making them useful for equation verification.
Do You Know?
- The principle of homogeneity states that only quantities with the same dimensions can be added or equated.
- Physical constants such as Planck’s constant and gravitational constant have specific dimensional formulas.
- Dimensional analysis can help derive unit conversions without memorization.
Worksheet
- Write the dimensional formula for pressure.
- Determine whether the equation v=u+atv = u + at is dimensionally correct.
- Find the dimensional formula for angular momentum.
- List three applications of dimensional analysis.
Test Paper (Marks Distribution)
Section A: Objective Type Questions (1 mark each)
- Write the dimensional formula of momentum.
- Which of the following has the same dimensions as force?
(a) Work (b) Power (c) Impulse (d) Pressure
Section B: Short Answer Questions (2 marks each)
- Explain why dimensions are useful in physics.
- Find the dimensional formula of power.
Section C: Long Answer Questions (5 marks each)
- Derive the dimensional formula of work and energy.
- How can dimensional analysis be used to check the correctness of an equation? Provide an example.
Important Points for Quick Revision
- Dimensions help in checking the correctness of equations.
- Dimensional formula represents the base quantities in a derived quantity.
- Work, energy, and torque have the same dimensional formula [ML²T⁻²].
- Dimensional analysis is useful for deriving relationships between physical quantities.
Test Your Knowledge(Quiz)
Dimensions of Physical Quantities Quiz
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