Physical Quantities and Their Dimensional Formula – Complete Guide
Complete NCERT Guide for CBSE, NEET, JEE, CUET and Competitive Examinations
Key Takeaways
- Understand the meaning and importance of physical quantities in Physics.
- Learn the difference between fundamental, derived and supplementary quantities.
- Master SI units and dimensional formulas with easy examples.
- Understand dimensional analysis and its practical applications.
- Includes important tables, exam tips, common mistakes and FAQs.
- Useful for NCERT, CBSE, NEET, JEE, CUET and other competitive examinations.
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| Physical Quantities and Their Dimensional Formula |
Introduction
Every measurable phenomenon in physics is described using a physical quantity. Whether we measure the length of a table, the mass of an object, the speed of a moving vehicle, or the force applied to a body, we are dealing with physical quantities. These quantities help scientists describe natural phenomena accurately using numbers and standard units.
A physical quantity consists of two essential parts: a numerical value (magnitude) and a unit. For example, when we say that the length of a rod is 5 metres, the number 5 represents the magnitude, while metre (m) represents the unit. Without standard units, scientific measurements would be inconsistent and impossible to compare across different regions of the world.
To simplify measurements, the International System of Units (SI System) defines seven fundamental physical quantities. All other quantities used in science and engineering are derived from these fundamental quantities through mathematical relationships. Examples of derived quantities include force, work, power, pressure, velocity, acceleration and density.
Closely related to physical quantities is the concept of the dimensional formula. A dimensional formula expresses a physical quantity in terms of the powers of the fundamental quantities such as mass (M), length (L) and time (T). Dimensional analysis helps verify equations, derive relationships between physical quantities and convert units from one system to another.
Understanding physical quantities and dimensional formulas forms the foundation of mechanics, thermodynamics, electricity, optics and modern physics. This topic is frequently tested in NCERT, CBSE Board Examinations, NEET, JEE Main & Advanced, CUET, SSC, UPSC and various State PSC examinations. A clear understanding of these concepts also makes solving numerical problems much easier.
Quick Fact
- Every physical quantity consists of a magnitude and a unit.
- The SI system contains seven fundamental physical quantities.
- Derived quantities are obtained by combining fundamental quantities.
- Dimensional formulas are used to check the correctness of physical equations.
- Dimensional analysis is one of the most important tools in Physics.
What is a Physical Quantity?
A physical quantity is any property of a substance, object or natural phenomenon that can be measured and expressed using a numerical value along with a standard unit. Every measurement in Physics is represented by a physical quantity. Whether measuring the height of a building, the speed of a car, the temperature of water or the force acting on an object, we are dealing with physical quantities.
A physical quantity always consists of two essential components:
Magnitude
The numerical value representing how much of a quantity is present.
Example: 25 in 25 kg.
Unit
The internationally accepted standard used for measurement.
Example: kilogram (kg).
If the length of a table is 2 metres, then:
- Magnitude = 2
- Unit = metre (m)
Classification of Physical Quantities
Physical quantities are classified into three major categories depending on their dependence on other quantities.
1. Fundamental Physical Quantities
Independent quantities that cannot be expressed in terms of any other physical quantities.
2. Derived Physical Quantities
Obtained by combining two or more fundamental quantities using mathematical relationships.
3. Supplementary Physical Quantities
Quantities having SI units but no dimensions.
Remember
Every derived quantity ultimately depends upon one or more of the seven SI base quantities.
Fundamental Physical Quantities
Fundamental physical quantities are the basic quantities on which the entire SI system is built. They are independent quantities and cannot be derived from any other physical quantity.
The International System of Units (SI) recognizes seven fundamental physical quantities.
| Fundamental Quantity | SI Unit | Symbol | Dimension |
|---|---|---|---|
| Length | metre | m | [L] |
| Mass | kilogram | kg | [M] |
| Time | second | s | [T] |
| Electric Current | ampere | A | [I] |
| Thermodynamic Temperature | kelvin | K | [Θ] |
| Amount of Substance | mole | mol | [N] |
| Luminous Intensity | candela | cd | [J] |
Characteristics of Fundamental Quantities
- They are independent of one another.
- They form the foundation of the SI unit system.
- Every derived quantity is expressed using these quantities.
- They are universally accepted for scientific measurements.
Exam Tip
Questions asking the SI unit, symbol, and dimensional symbol of the seven base quantities are frequently asked in CBSE Board, NEET, JEE Main, CUET, SSC and various competitive examinations.
Common Mistakes
- Writing gram (g) instead of kilogram (kg) as the SI base unit of mass.
- Confusing quantity symbols with unit symbols.
- Using Celsius instead of Kelvin as the SI base unit of temperature.
- Assuming force and energy are fundamental quantities—they are derived quantities.
Derived Physical Quantities
Derived physical quantities are quantities that are obtained by combining two or more fundamental physical quantities through mathematical operations such as multiplication or division. Unlike fundamental quantities, derived quantities are dependent on the SI base quantities and can always be expressed in terms of their dimensional formulas.
Almost every quantity used in mechanics, thermodynamics, electricity, optics and modern physics is a derived quantity. Examples include velocity, acceleration, force, pressure, work, energy and power.
Velocity is obtained by dividing displacement by time.
Velocity = Distance ÷ Time
SI Unit = metre per second (m/s)
Dimensional Formula = [LT-1]
Common Derived Physical Quantities
| Physical Quantity | SI Unit | Dimensional Formula |
|---|---|---|
| Area | m² | [L²] |
| Volume | m³ | [L³] |
| Velocity | m/s | [LT-1] |
| Acceleration | m/s² | [LT-2] |
| Momentum | kg m/s | [MLT-1] |
| Force | Newton (N) | [MLT-2] |
| Work | Joule (J) | [ML²T-2] |
| Energy | Joule (J) | [ML²T-2] |
| Power | Watt (W) | [ML²T-3] |
| Pressure | Pascal (Pa) | [ML-1T-2] |
| Density | kg/m³ | [ML-3] |
| Frequency | Hertz (Hz) | [T-1] |
| Impulse | N s | [MLT-1] |
| Surface Tension | N/m | [MT-2] |
| Gravitational Constant | N m²/kg² | [M-1L³T-2] |
Quick Memory Trick
- Velocity → Distance ÷ Time
- Acceleration → Velocity ÷ Time
- Force → Mass × Acceleration
- Pressure → Force ÷ Area
- Power → Work ÷ Time
Supplementary Physical Quantities
Supplementary physical quantities are special quantities that possess SI units but are considered dimensionless because they represent ratios of similar physical quantities. Although the SI system now treats them as derived quantities with the dimension one, they are still commonly discussed in introductory physics.
| Quantity | SI Unit | Symbol | Dimension |
|---|---|---|---|
| Plane Angle | Radian | rad | Dimensionless |
| Solid Angle | Steradian | sr | Dimensionless |
Characteristics of Supplementary Quantities
- Represent angular measurements.
- Have SI units but no independent dimensions.
- Widely used in geometry, mechanics and optics.
- Frequently used in rotational motion and wave optics.
Important Note
According to the modern SI system, radian and steradian are treated as dimensionless derived units. However, they are still included in many school and competitive examination syllabi under supplementary quantities.
Real-Life Applications
- Velocity is used to measure vehicle speed.
- Pressure is important in hydraulic systems.
- Energy is measured in electrical appliances.
- Power determines the efficiency of machines.
- Frequency is widely used in sound waves and radio communication.
- Angular measurements are essential in satellite technology and astronomy.
What is Dimensional Formula?
A dimensional formula is a mathematical representation of a physical quantity in terms of the powers of the fundamental quantities such as Mass (M), Length (L), Time (T), Electric Current (I), Temperature (Θ), Amount of Substance (N) and Luminous Intensity (J).
Every derived physical quantity can be expressed as a combination of these fundamental dimensions. Dimensional formulas help physicists understand how different quantities are related without considering numerical constants.
General Representation
A dimensional formula is generally written as:
[MaLbTcIdΘeNfJg]
where a, b, c, d, e, f and g are the powers of the corresponding base quantities.
Fundamental Dimensions
| Symbol | Fundamental Quantity |
|---|---|
| M | Mass |
| L | Length |
| T | Time |
| I | Electric Current |
| Θ | Temperature |
| N | Amount of Substance |
| J | Luminous Intensity |
Examples of Important Dimensional Formulae
The following dimensional formulas are among the most frequently asked in school examinations and competitive tests such as NEET, JEE, CUET and SSC.
| Physical Quantity | Dimensional Formula |
|---|---|
| Velocity | [LT-1] |
| Acceleration | [LT-2] |
| Momentum | [MLT-1] |
| Force | [MLT-2] |
| Impulse | [MLT-1] |
| Work | [ML2T-2] |
| Energy | [ML2T-2] |
| Power | [ML2T-3] |
| Pressure | [ML-1T-2] |
| Density | [ML-3] |
| Frequency | [T-1] |
| Surface Tension | [MT-2] |
| Young's Modulus | [ML-1T-2] |
| Gravitational Constant | [M-1L3T-2] |
| Planck's Constant | [ML2T-1] |
| Electric Charge | [IT] |
| Potential Difference | [ML2T-3I-1] |
| Resistance | [ML2T-3I-2] |
| Capacitance | [M-1L-2T4I2] |
| Magnetic Flux | [ML2T-2I-1] |
Importance of Dimensional Analysis
Dimensional analysis is one of the most powerful tools in Physics. It allows scientists and students to verify equations, derive relationships and convert units without performing lengthy calculations.
Checking Equations
Ensures that both sides of a physical equation have identical dimensions.
Unit Conversion
Helps convert measurements from one system of units to another accurately.
Deriving Formulae
Useful for establishing relationships between physical quantities.
Error Detection
Incorrect equations can often be identified by comparing dimensions.
Applications of Dimensional Analysis
- Verification of physical equations.
- Conversion between SI, CGS and FPS systems.
- Finding unknown relationships among variables.
- Checking the dimensional consistency of derived equations.
- Solving engineering and scientific problems.
- Reducing calculation errors during numerical analysis.
Exam Tip
Questions on dimensional formulas of force, pressure, power, density, gravitational constant, Planck's constant and electric quantities are among the most frequently asked in competitive examinations.
Limitations of Dimensional Analysis
- Cannot determine numerical constants such as 2, π or ½.
- Cannot distinguish between quantities having identical dimensions.
- Cannot derive equations involving trigonometric, exponential or logarithmic functions.
- Only verifies dimensional correctness, not the complete correctness of an equation.
Summary
Physical quantities form the foundation of Physics because every measurable phenomenon is expressed using them. A physical quantity always consists of a numerical value and a standard unit. Based on their dependence on other quantities, physical quantities are classified into fundamental, derived, and supplementary quantities.
The SI system recognizes seven fundamental quantities, while all other quantities are derived from them. Dimensional formulas express physical quantities in terms of fundamental dimensions and play an essential role in checking equations, converting units and solving physics problems. A strong understanding of dimensional analysis greatly improves problem-solving skills in both academic and competitive examinations.
Quick Revision
- Every physical quantity consists of a magnitude and a unit.
- There are seven SI fundamental physical quantities.
- Derived quantities are combinations of fundamental quantities.
- Plane angle and solid angle are dimensionless quantities with SI units.
- Dimensional formulas express quantities in terms of M, L, T, I, Θ, N and J.
- Dimensional analysis helps verify equations and convert units.
- Force → [MLT-2]
- Energy → [ML2T-2]
- Pressure → [ML-1T-2]
- Power → [ML2T-3]
Common Mistakes Students Make
- Confusing fundamental quantities with derived quantities.
- Using gram (g) instead of kilogram (kg) as the SI base unit of mass.
- Writing incorrect signs in dimensional powers.
- Confusing SI units with dimensional formulas.
- Assuming dimensional analysis can derive every equation.
- Ignoring unit consistency while solving numerical problems.
Competitive Examination Points
CBSE Boards
Learn all seven SI base quantities, SI units and common dimensional formulas.
NEET
Focus on dimensional analysis, unit conversion and dimensional formulas of mechanics.
JEE Main & Advanced
Practice derivation of dimensional formulas and application-based numerical problems.
CUET & SSC
Prepare frequently asked objective questions related to SI units and dimensions.
Frequently Asked Questions (FAQs)
1. What is a physical quantity?
A physical quantity is any measurable property of a body or phenomenon that can be expressed by a numerical value along with a standard unit.
2. How many SI fundamental physical quantities are there?
There are seven SI fundamental physical quantities: Length, Mass, Time, Electric Current, Thermodynamic Temperature, Amount of Substance and Luminous Intensity.
3. What is a derived physical quantity?
A derived physical quantity is obtained by combining two or more fundamental quantities through mathematical operations such as multiplication or division.
4. What is the dimensional formula of force?
The dimensional formula of force is [MLT-2].
5. What is the dimensional formula of energy?
The dimensional formula of energy is [ML2T-2].
6. Why is dimensional analysis important?
Dimensional analysis helps verify physical equations, convert units, derive relationships among physical quantities and identify dimensional errors in equations.
7. Can dimensional analysis derive every equation?
No. Dimensional analysis cannot determine numerical constants such as π, 2 or ½ and cannot derive equations involving trigonometric, exponential or logarithmic functions.
8. What are supplementary physical quantities?
Plane angle (radian) and solid angle (steradian) are traditionally known as supplementary quantities. In the modern SI system they are treated as dimensionless derived quantities.
9. Which dimensional formulas are most important for competitive examinations?
Force, Work, Energy, Power, Pressure, Density, Momentum, Frequency, Gravitational Constant, Planck's Constant, Electric Charge and Potential Difference are among the most frequently asked.
10. Which examinations include questions on dimensional formulas?
Questions from this topic are commonly asked in CBSE, ICSE, NEET, JEE Main & Advanced, CUET, SSC, Railway, UPSC, State PSC and other competitive examinations.
Conclusion
Physical quantities and dimensional formulas form the foundation of Physics because every measurable phenomenon is expressed using standard units and dimensions. A clear understanding of SI units, fundamental and derived quantities, and dimensional analysis enables students to solve numerical problems accurately and verify the correctness of physical equations.
Regular practice of dimensional formulas and SI units not only strengthens conceptual understanding but also improves performance in school examinations and competitive examinations such as NEET, JEE, CUET and SSC.
Final Revision
- ✔ Every physical quantity consists of a magnitude and a unit.
- ✔ There are seven SI base quantities.
- ✔ Derived quantities are obtained from fundamental quantities.
- ✔ Dimensional formulas represent quantities using M, L, T, I, Θ, N and J.
- ✔ Dimensional analysis verifies equations and converts units.
- ✔ Force → [MLT-2]
- ✔ Energy → [ML2T-2]
- ✔ Pressure → [ML-1T-2]
- ✔ Power → [ML2T-3]
References
- NCERT Physics Class 11
- NCERT Exemplar Problems – Physics
- Fundamentals of Physics – Halliday, Resnick & Walker
- Concepts of Physics – H. C. Verma
- SI Brochure – Bureau International des Poids et Mesures (BIPM)