UNIT AND DIMENSIONS
UNIT AND DIMENSIONS
Study Notes
UNIT AND DIMENSIONS
Complete Physics Notes – Class 11 | IIT-JEE Main & Advanced
1. Introduction
Physics is a subject in which we study nature and the laws that explain different physical phenomena. During the study of physics, we measure many quantities such as length, mass, time, velocity, force and energy.
For example, if we say that a table is 2 metres long, we are expressing its length using a numerical value and a unit.
A physical measurement can be written as:
Physical quantity = Numerical value × Unit
Here, the numerical value tells us how many times the unit is contained in the measured quantity, while the unit provides the standard of comparison.
Units and dimensions are important because they help us understand physical quantities, check equations, convert units and derive relationships between different quantities.
2. Physical Quantities
A physical quantity is a property that can be measured and expressed numerically with an appropriate unit.
Examples include length, mass, time, temperature, speed, acceleration, force and pressure.
Physical quantities are classified into two main types.
A. Fundamental Quantities
Fundamental quantities are independent quantities that are not defined in terms of other physical quantities.
The seven fundamental quantities in the SI system are:
| Quantity | SI Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
These quantities form the foundation of the SI system.
B. Derived Quantities
Derived quantities are obtained by combining fundamental quantities mathematically.
Examples:
Area = Length × Breadth
Volume = Length × Breadth × Height
Speed = Distance / Time
Acceleration = Change in velocity / Time
Force = Mass × Acceleration
Density = Mass / Volume
Pressure = Force / Area
For example, the SI unit of speed is m/s, and the SI unit of force is kg m/s², which is called a newton.
3. Systems of Units
Different systems of units have been used for measuring physical quantities.
CGS System
Length: centimetre
Mass: gram
Time: second
MKS System
Length: metre
Mass: kilogram
Time: second
FPS System
Length: foot
Mass: pound
Time: second
SI System
SI stands for International System of Units. It is the internationally accepted system used in scientific measurements.
Some important derived units are:
| Quantity | SI Unit | Symbol |
|---|---|---|
| Force | newton | N |
| Work | joule | J |
| Energy | joule | J |
| Power | watt | W |
| Pressure | pascal | Pa |
| Frequency | hertz | Hz |
| Electric charge | coulomb | C |
| Potential difference | volt | V |
4. Iimportant Unit Conversions
Some commonly used conversions are:
1 km = 10³ m
1 m = 10² cm
1 cm = 10⁻² m
1 mm = 10⁻³ m
1 μm = 10⁻⁶ m
1 nm = 10⁻⁹ m
1 Å = 10⁻¹⁰ m
1 kg = 10³ g
1 g = 10⁻³ kg
1 hour = 3600 s
1 day = 86400 s
1 litre = 10⁻³ m³
1 mL = 1 cm³
Conversion of Speed
To convert km/h into m/s:
Multiply by 5/18.
To convert m/s into km/h:
Multiply by 18/5.
Example:
Convert 72 km/h into m/s.
Solution:
72 × 5/18 = 20 m/s.
Therefore, the answer is 20 m/s.
5. Meaning of Dimensions
Dimensions describe how a physical quantity depends on fundamental quantities.
In mechanics, we commonly use:
M for mass
L for length
T for time
For example, velocity is defined as displacement divided by time.
Velocity = Displacement / Time
Therefore:
[Velocity] = [LT⁻¹]
Similarly:
Acceleration = Velocity / Time
[Acceleration] = [LT⁻²]
Dimensions describe the physical nature of a quantity, but they do not provide its numerical value.
6. Dimensional Formula
A dimensional formula expresses a physical quantity in terms of the powers of fundamental quantities.
The general form is:
[MᵃLᵇTᶜ]
For quantities involving electricity and temperature, additional fundamental dimensions may be required.
Example 1: Force
According to Newton's second law:
Force = Mass × Acceleration
Dimensions of mass = [M]
Dimensions of acceleration = [LT⁻²]
Therefore:
[F] = [M] × [LT⁻²]
[F] = [MLT⁻²]
Example 2: Momentum
Momentum = Mass × Velocity
[p] = [M] × [LT⁻¹]
[p] = [MLT⁻¹]
Example 3: Work
Work = Force × Displacement
[W] = [MLT⁻²] × [L]
[W] = [ML²T⁻²]
Example 4: Power
Power = Work / Time
[P] = [ML²T⁻²]/[T]
[P] = [ML²T⁻³]
7. Important Dimensional Formulae
These dimensional formulas should be memorised for IIT-JEE.
| Physical Quantity | Dimensional Formula |
|---|---|
| Area | [L²] |
| Volume | [L³] |
| Density | [ML⁻³] |
| 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] |
| Stress | [ML⁻¹T⁻²] |
| Strain | [M⁰L⁰T⁰] |
| Frequency | [T⁻¹] |
| Time period | [T] |
| Angular velocity | [T⁻¹] |
| Angular acceleration | [T⁻²] |
| Torque | [ML²T⁻²] |
| Angular momentum | [ML²T⁻¹] |
| Spring constant | [MT⁻²] |
| Gravitational constant | [M⁻¹L³T⁻²] |
| Surface tension | [MT⁻²] |
| Dynamic viscosity | [ML⁻¹T⁻¹] |
| Kinematic viscosity | [L²T⁻¹] |
| Planck's constant | [ML²T⁻¹] |
Electrical Quantities
| Quantity | Dimensional Formula |
|---|---|
| Electric charge | [IT] |
| Potential difference | [ML²T⁻³I⁻¹] |
| Resistance | [ML²T⁻³I⁻²] |
| Capacitance | [M⁻¹L⁻²T⁴I²] |
| Magnetic field | [MT⁻²I⁻¹] |
Important observation: Work and torque have the same dimensions, but they represent different physical quantities. Dimensional formulas alone cannot distinguish between all quantities.
8. Dimensionless Quantities
Dimensionless quantities have no net dimensions.
Their dimensional formula is:
[M⁰L⁰T⁰] = [1]
Examples include:
Strain
Refractive index
Relative density
Coefficient of friction
Angle in radians
Pure numbers such as π
Example: Angle
Angle = Arc length / Radius
Both arc length and radius have dimensions of length.
Therefore:
[Angle] = [L]/[L] = [1]
Hence, angle is dimensionless.
A dimensionless quantity can still have a named SI unit. For example, the radian is the named unit for plane angle.
9. Principle of Homogeneity of Dimensions
According to the principle of homogeneity, all terms added or subtracted in a physically meaningful equation must have the same dimensions.
Consider:
s = ut + ½at²
Here:
s = displacement
u = initial velocity
t = time
a = acceleration
Dimensions of the left-hand side:
[s] = [L]
Dimensions of ut:
[ut] = [LT⁻¹][T] = [L]
Dimensions of at²:
[at²] = [LT⁻²][T²] = [L]
All terms have dimensions of length.
Therefore, the equation is dimensionally homogeneous.
Important: A dimensionally homogeneous equation is not necessarily physically correct. Dimensional analysis provides a necessary check, not complete proof.
10. Applications of Dimensional Analysis
Dimensional analysis is useful in checking equations, deriving relationships and converting units.
A. Checking the Correctness of Equations
Consider:
v² = u² + 2as
Dimensions of v² = [L²T⁻²]
Dimensions of u² = [L²T⁻²]
Dimensions of as:
[LT⁻²] × [L] = [L²T⁻²]
All terms have the same dimensions.
Therefore, the equation is dimensionally homogeneous.
B. Deriving Relations Between Physical Quantities
Suppose the time period T of a simple pendulum depends on its length l and gravitational acceleration g.
Assume:
T ∝ lᵃgᵇ
Therefore:
T = klᵃgᵇ
where k is a dimensionless constant.
Substituting dimensions:
[T] = [L]ᵃ[LT⁻²]ᵇ
[T] = [Lᵃ⁺ᵇT⁻²ᵇ]
Comparing the powers of time:
1 = −2b
Therefore:
b = −1/2
Comparing the powers of length:
0 = a + b
Therefore:
a = 1/2
Hence:
T = k√(l/g)
For a simple pendulum undergoing small oscillations, the actual expression is:
T = 2π√(l/g)
Dimensional analysis determines the powers of l and g, but it cannot determine the numerical constant 2π.
11. Limitations of Dimensional Analysis
Dimensional analysis has some important limitations:
It cannot determine dimensionless numerical constants such as 2, ½ or 2π.
It cannot distinguish between physical quantities having the same dimensions.
It cannot determine whether terms should be added or subtracted.
It cannot independently derive the complete form of equations involving trigonometric, exponential or logarithmic functions.
A dimensionally correct equation may still be physically incorrect.
For example, both equations below are dimensionally homogeneous:
s = ut + at²
s = ut + ½at²
Dimensional analysis alone cannot establish which equation correctly describes a particular physical situation.
12. Significant Figures
Significant figures are the meaningful digits in a measured value, including the final uncertain digit.
They indicate the precision of a measurement.
Rules of Significant Figures
Rule 1: All non-zero digits are significant.
Example: 245 has 3 significant figures.
Rule 2: Zeros between non-zero digits are significant.
Example: 2005 has 4 significant figures.
Rule 3: Leading zeros are not significant.
Example: 0.0045 has 2 significant figures.
Rule 4: Trailing zeros after a decimal point are significant.
Example: 2.500 has 4 significant figures.
Rule 5: Trailing zeros in a whole number without a decimal point can be ambiguous.
For example, 1500 may represent 2, 3 or 4 significant figures depending on the intended precision.
Scientific notation removes this ambiguity:
1.5 × 10³ has 2 significant figures.
1.500 × 10³ has 4 significant figures.
Operations With Significant Figures
Addition and subtraction: Round the result to the least precise number of decimal places.
Example:
12.4 + 3.26 = 15.66
Rounded to one decimal place:
15.7
Multiplication and division: Round the result to the same number of significant figures as the input with the fewest significant figures.
Example:
2.5 × 3.42 = 8.55
Rounded to 2 significant figures:
8.6
13. Accuracy, Precision and Errors
Accuracy describes how close a measured value is to the true or accepted value.
Precision describes how close repeated measurements are to one another.
A measurement can be precise without being accurate.
Absolute Error
For a measured value aᵢ and the mean value ā:
Absolute error = |aᵢ − ā|
For n measurements:
ā = (a₁ + a₂ + ... + aₙ)/n
The mean absolute error is:
Δā = Σ|aᵢ − ā|/n
Relative Error
Relative error = Δā/|ā|
Percentage Error
Percentage error = (Δā/|ā|) × 100%
Error Propagation
If:
Z = A × B
then the approximate maximum fractional error is:
ΔZ/Z = ΔA/A + ΔB/B
If:
Z = A/B
then:
ΔZ/Z =ΔA/A + ΔB/B
If:
Z = Aⁿ
then:
ΔZ/Z ≈ |n|ΔA/A
These are standard first-order approximations for small uncertainties.
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14. Vernier Caliperss
Vernier calipers are used to measure external diameters, internal diameters, lengths and depths.
Least Count
For a standard direct vernier:
Least count = 1 main scale division − 1 vernier scale division
Reading Formula
Observed reading = Main scale reading + (Vernier coincidence × Least count)
If zero error is present:
Correct reading = Observed reading − Zero error
The sign of the zero error must be handled carefully.
15. Screw Gaugees
A screw gauge is used to measure very small dimensions, such as the diameter of a thin wire.
Pitch
Pitch is the distance moved by the screw in one complete rotation.
Least Count
Least count = Pitch / Number of circular scale divisions
Reading Formula
Observed reading = Pitch scale reading + (Circular scale reading × Least count)
Correct reading = Observed reading − Zero error
16. Important Solved Examples
Example 1: Find the dimensional formula of pressure.
Pressure = Force / Area
[P] = [MLT⁻²]/[L²]
Answer: [ML⁻¹T⁻²]
Example 2: Find the dimensional formula of the gravitational constant G.
Newton's law of gravitation:
F = Gm₁m₂/r²
Therefore:
G = Fr²/(m₁m₂)
[G] = [MLT⁻²][L²]/[M²]
Answer: [M⁻¹L³T⁻²]
Example 3: Find the dimensional formula of the spring constant.
From Hooke's law:
F = kx
Therefore:
k = F/x
[k] = [MLT⁻²]/[L]
Answer: [MT⁻²]
Example 4: Find the percentage error in the volume of a sphere if the percentage error in its radius is 2%.
Volume:
V = (4/3)πr³
Therefore:
ΔV/V ≈ 3Δr/r
Percentage error in volume = 3 × 2%
Answer: 6%
Example 5: Convert 36 km/h into m/s.
36 × 5/18 = 10 m/s
Answer: 10 m/s
17. Practice Questions for IIT-JEEes
Try solving these questions without looking at the answers.
Find the dimensional formula of momentum.
Find the dimensional formula of pressure.
Check whether v = u + at is dimensionally homogeneous.
Find the dimensional formula of gravitational constant G.
If the percentage error in the radius of a sphere is 3%, find the approximate percentage error in its volume.
How many significant figures are present in 0.005060?
Find the dimensional formula of spring constant k.
Convert 72 km/h into m/s.
Which physical quantities have the same dimensions as work?
State two limitations of dimensional analysis.
Answer Key
[MLT⁻¹]
[ML⁻¹T⁻²]
Yes
[M⁻¹L³T⁻²]
9%
4
[MT⁻²]
20 m/s
Energy and torque have the same dimensions as work.
It cannot determine dimensionless constants or distinguish between quantities with identical dimensions.
[MLT⁻¹]
[ML⁻¹T⁻²]
Yes
[M⁻¹L³T⁻²]
9%
4
[MT⁻²]
20 m/s
Energy and torque have the same dimensions as work.
It cannot determine dimensionless constants or distinguish between quantities with identical dimensions.
18. Quick Revision
Before attempting JEE questions, remember these important formulas:
Velocity = [LT⁻¹]
Acceleration = [LT⁻²]
Force = [MLT⁻²]
Momentum = [MLT⁻¹]
Work and Energy = [ML²T⁻²]
Power = [ML²T⁻³]
Pressure = [ML⁻¹T⁻²]
Density = [ML⁻³]
Gravitational constant = [M⁻¹L³T⁻²]
Spring constant = [MT⁻²]
Frequency = [T⁻¹]
Angular momentum = [ML²T⁻¹]
thank you so much jee crakers 👍👍✔✔ by shivam shreevash
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