EDUNES ONLINE EDUCATION: Dimensions of Physical Quantities

Edunes Online Education

Dimensions of Physical Quantities from Section 1.4 – CBSE Class 11 Physics (Units and Measurements)

📘 1.4 – Dimensions of Physical Quantities

🔍 What are Dimensions?

The dimension of a physical quantity refers to the power (or exponent) to which a base quantity must be raised to represent that physical quantity.

For any derived quantity, we express it in terms of fundamental dimensions like:

\text{[Length]} = [L], \quad \text{[Mass]} = [M], \quad \text{[Time]} = [T], \quad \text{[Electric Current]} = [A], \\ \text{[Temperature]} = [K], \quad \text{[Luminous Intensity]} = [cd], \quad \text{[Amount of Substance]} = [mol]

These seven are the fundamental or base dimensions in physics.

🔢 Representing Dimensions

The notation [Q] refers to the dimensions of a physical quantity Q.

For example:

If $Q = \text{Force}$ , then $[Q] = [M L T^{-2}]$
If $Q = \text{Speed}$ , then $[Q] = [L T^{-1}]$

📐 Examples in Mechanics

In mechanics, all quantities can be expressed using only [M], [L], [T].

📦 Example 1: Volume

Volume = Length × Breadth × Height
Each is of dimension $[L]$ , so:

[\text{Volume}] = [L] \cdot [L] \cdot [L] = [L^3]

It has:

3 dimensions in length
0 in mass $[M^0]$
0 in time $[T^0]$

🧲 Example 2: Force

\text{Force} = \text{Mass} \times \text{Acceleration}

[\text{Mass}] = [M], \quad [\text{Acceleration}] = \frac{[L]}{[T^2]} = [L T^{-2}]

[\text{Force}] = [M] \cdot [L T^{-2}] = [M L T^{-2}]

Force has:

1 dimension in mass
1 in length
–2 in time

🚗 Example 3: Velocity

Velocity = Displacement / Time

[\text{Velocity}] = \frac{[L]}{[T]} = [L T^{-1}]

Same applies to:

Initial velocity
Final velocity
Average velocity
Speed

Even though magnitudes vary, their dimensions remain the same.

✅ Important Characteristics of Dimensional Representation

Only the type or nature of quantity is described.
It does not involve magnitude or unit.
Different quantities with same dimensional formula are called dimensionally similar.
It helps:
- Check dimensional consistency of equations
- Derive relations between quantities
- Convert units between systems

🧮 Zero Dimensions in a Base Quantity

A quantity can have zero power in a particular base quantity, meaning it doesn't depend on it.

For example:

Volume: $[M^0 L^3 T^0]$
Speed: $[M^0 L^1 T^{-1}]$

🚫 Not All Physical Properties Are Dimensioned

Quantities like refractive index, strain, angle (in radians) are dimensionless.
Still physical, but have no associated base dimension.

🧠 Summary Table

Physical Quantity	Dimensional Formula
Speed/Velocity	$[L T^{-1}]$
Acceleration	$[L T^{-2}]$
Force	$[M L T^{-2}]$
Work/Energy	$[M L^2 T^{-2}]$
Power	$[M L^2 T^{-3}]$
Pressure	$[M L^{-1} T^{-2}]$
Density	$[M L^{-3}]$
Momentum	$[M L T^{-1}]$

🧾 Conclusion

Understanding dimensions is essential for:

Analyzing physical relationships
Verifying equations
Deriving new formulas
Performing conversions across unit systems

By using dimensional analysis, we gain powerful insight into the structure of physical laws, without the need for detailed experiments at every step.

Based on the principles of Units and Measurements discussed in the Dimensions of Physical Quantities article, physical quantities can be classified into four distinct categories based on whether they have fixed values and whether they possess dimensions.

1. Dimensional Variables

These are physical quantities that possess dimensions and do not have a fixed value (their value changes depending on the situation). Most quantities in mechanics fall into this category.

Characteristics: They have a dimensional formula and vary in magnitude.
Examples:
- Velocity: $[LT^{-1}]$
- Force: $[MLT^{-2}]$
- Work/Energy: $[ML^2T^{-2}]$
- Power: $[ML^2T^{-3}]$

2. Dimensionless Variables

These are physical quantities that have variable values but no dimensions. These usually arise when a quantity is a ratio of two similar physical quantities.

Characteristics: They are represented as $[M^0L^0T^0]$ and vary in magnitude.
Examples:
- Angle: Ratio of arc length to radius.
- Strain: Ratio of change in dimension to original dimension.
- Refractive Index: Ratio of the speed of light in two different media.
- Relative Density: Ratio of the density of a substance to the density of water.

3. Dimensional Constants

These are physical quantities that have fixed values (constants) but still possess dimensions. These are fundamental constants used in physical laws.

Characteristics: They have a specific numerical value and a dimensional formula.
Examples:
- Gravitational Constant ($G$): $[M^{-1}L^3T^{-2}]$ (Value: $6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$)
- Planck’s Constant ($h$): $[ML^2T^{-1}]$
- Velocity of Light ($c$): $[LT^{-1}]$
- Gas Constant ($R$): $[ML^2T^{-2}K^{-1}\text{mol}^{-1}]$

4. Dimensionless Constants

These are pure numbers or mathematical constants that have fixed values and no dimensions.

Characteristics: They are numerical values with no units or dimensions ($[M^0L^0T^0]$).
Examples:
- Pure Numbers: $1, 2, 3, \dots, 100$.
- Mathematical Constants: $\pi$ (approx. $3.14$) and $e$ (approx. $2.71$).
- Trigonometric Ratios: $\sin \theta, \cos \theta$, etc. (since they are ratios of sides of a triangle).

Summary Table for Quick Revision

Category	Dimensions?	Value?	Examples
Dimensional Variables	Yes	Variable	Force, Velocity, Area
Dimensionless Variables	No	Variable	Strain, Angle, Specific Gravity
Dimensional Constants	Yes	Constant	$G$, $h$, $c$, Stefan's Constant
Dimensionless Constants	No	Constant	$\pi$, $e$, Pure numbers (5, 10)

Quick Tip for NEET/JEE: If you see a ratio of the same two quantities (like speed/speed or length/length), it is almost always a Dimensionless Variable. If you see a fundamental constant from a formula (like $G$ in Newton's Law of Gravitation), it is a Dimensional Constant.

Based on the Dimensions of Physical Quantities article, here are NEET-standard Multiple Choice Questions (MCQs) focusing on conceptual depth and dimensional analysis applications.

NEET Practice Questions: Dimensions

1. Which of the following pairs of physical quantities have the same dimensional formula?

A. Force and Power

B. Work and Energy

C. Velocity and Acceleration

D. Pressure and Force

2. The dimensional formula for Kinetic Energy is given by:

A. $[MLT^{-2}]$

B. $[ML^2T^{-2}]$

C. $[ML^2T^{-3}]$

D. $[ML^{-1}T^{-2}]$

3. A physical quantity $X$ is calculated by the formula $X = \frac{\text{Force}}{\text{Area}}$. The dimensions of $X$ are:

A. $[ML^{-1}T^{-2}]$

B. $[ML^2T^{-2}]$

C. $[MLT^{-2}]$

D. $[M^0L^0T^0]$

4. Which of the following is a dimensionless quantity?

A. Refractive Index

B. Density

C. Momentum

D. Specific Heat

5. If the dimension of a quantity is $[ML^2T^{-3}]$, the quantity is:

A. Work

B. Power

C. Pressure

D. Force

Answer Key & Explanations

Q. No	Answer	Explanation
1	B	Both Work and Energy have the dimensional formula $[ML^2T^{-2}]$.
2	B	Energy (any form) involves Work done, which is Force × Distance: $[MLT^{-2}] \times [L] = [ML^2T^{-2}]$.
3	A	$X$ represents Pressure. $\frac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}]$.
4	A	Refractive Index is a ratio of similar quantities (speeds), making it dimensionless $[M^0L^0T^0]$.
5	B	Power is Work/Time. $\frac{[ML^2T^{-2}]}{[T]} = [ML^2T^{-3}]$.

Quick Revision Table

Quantity	Formula	Dimensions
Density	Mass / Volume	$[ML^{-3}]$
Momentum	Mass × Velocity	$[MLT^{-1}]$
Acceleration	Velocity / Time	$[LT^{-2}]$
Pressure	Force / Area	$[ML^{-1}T^{-2}]$

Note: In NEET, always remember that quantities like Strain, Angle, and Refractive Index are frequently tested because they have no dimensions.

This worksheet is designed based on the NEP (National Education Policy) guidelines for CBSE Class 11 Physics, focusing on Competency-Based Questions and Higher Order Thinking Skills (HOTS). It moves beyond rote memorization to test your ability to apply dimensional analysis to unfamiliar or complex scenarios.

HOTS Worksheet: Dimensions of Physical Quantities

Section A: Assertion & Reasoning (Competency-Based)

Directions: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:

(a) Both A and R are true, and R is the correct explanation of A.

(b) Both A and R are true, but R is NOT the correct explanation of A.

(c) A is true but R is false.

(d) A is false but R is true.

1. Assertion (A): A dimensionally correct equation may not be a physically correct equation.

Reason (R): Dimensionless constants cannot be determined through dimensional analysis.

2. Assertion (A): The dimensions of physical quantities like Angle and Strain are the same.

Reason (R): Both are dimensionless variables.

Section B: Analytical Case Study

Context: In a hypothetical universe, the fundamental quantities are chosen to be Force (F), Acceleration (A), and Time (T) instead of Mass, Length, and Time.

3. Application Task: Find the dimensional formula for Energy in terms of $F$, $A$, and $T$.

HINT: Use the method of Dimensional Analysis and Its Applications to set up the equation $E = k F^a A^b T^c$.

Section C: Critical Thinking & Problem Solving

4. The "Dimensionless" Paradox: The formula for the period of a simple pendulum is $T = 2\pi \sqrt{l/g}$.

Question: If a student erroneously derives the formula as $T = 2\pi(l/g)$, explain using the Dimensional Consistency of Equations why this formula is impossible, even if the student argues the constant $2\pi$ is correct.

5. Variable Analysis:

The Van der Waals equation for a real gas is:

$$\left( P + \frac{a}{V^2} \right) (V - b) = RT$$

Where $P$ is pressure and $V$ is volume. Determine the dimensions of the Dimensional Constants $a$ and $b$.

Apply the Principle of Homogeneity.

Section D: Answer Key & Evaluative Feedback

Q. No	Correct Answer	Self-Evaluation Insight
1	(a)	Remember: $s = ut + at^2$ is dimensionally correct but physically wrong (missing $1/2$).
2	(a)	Both are $[M^0L^0T^0]$. Identifying "dimensionless" pairs is a high-frequency NEET/JEE skill.
3	$[F A T^2]$	Energy = Work = Force $\times$ Displacement. Displacement = $1/2 \text{ (Acc.)} \times \text{ (Time)}^2$.
4	LHS $\neq$ RHS	LHS is $[T]$. RHS would be $[L] / [LT^{-2}] = [T^2]$. Units must match for physical reality.
5	$a: [ML^5T^{-2}]$; $b: [L^3]$	You can only add/subtract quantities with the same dimensions. $b$ must match Volume ($V$).

Reflective Summary for Your Portfolio

Dimensional Variables: Quantities like Force and Velocity that change and have units.
Dimensional Constants: Universal values like $G$ or $h$ that have units.
Dimensionless Constants: Pure numbers like $\pi$ or $e$.

Expert Challenge: Can you find a physical quantity that has units but no dimensions?

(Answer: Angle - it is measured in Radians/Degrees but has no dimensional formula!)

Units and Measurement: Part -1

Significant Numbers

Arithmatic Oprations with Significant Numbers

Rounding Off Uncertain Digits

Rules on Determining The Uncertainty In the results of Arithmatic Calculations

Dimesion of Physical Quantities

Dimensional Formula and Dimensional Equation

Worksheet: Dimensional Formula and Equations

Dimensional Consistency of Equations

Dimensional Analysis and Its Applications

Sample Paper on Units and Measurement

Answer Key- Deducing Relations among Physical Quantities