Units and Dimensions Short Notes for Exams

UNITS AND DIMENSIONS — SHORT NOTES

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1. Physical Quantities

Physical quantities are classified into:

• Fundamental (Base) quantities: Cannot be derived (e.g., length, mass, time, temperature, current, luminous intensity, amount of substance).

• Derived quantities: Formed from base quantities (e.g., velocity, force, energy, pressure).

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2. Units

(A) System of Units

1. CGS → cm, g, s

2. FPS → ft, pound, second

3. MKS/SI → metre, kg, second (Standard)

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(B) SI Base Units

Quantity	Symbol	SI Unit
Length	L	metre (m)
Mass	M	kilogram (kg)
Time	T	second (s)
Electric current	I	ampere (A)
Temperature	Θ	kelvin (K)
Amount of substance	n	mole (mol)
Luminous intensity	Iv	candela (cd)

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(C) Supplementary Units

• Plane angle → radian (rad)

• Solid angle → steradian (sr)

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3. Dimensions

Dimensions show how a quantity is built from base quantities.

Example:

• Velocity = distance/time → [LT⁻¹]

• Force = mass × acceleration → [MLT⁻²]

General form:

$$[M^a{L}^b{T}^c{I}^d{\mathrm{\Theta }}^e]$$

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4. Dimension of Common Physical Quantities (Must-Learn for Exams)

Quantity	Dimension
Velocity	[LT⁻¹]
Acceleration	[LT⁻²]
Momentum	[MLT⁻¹]
Force	[MLT⁻²]
Work/Energy	[ML²T⁻²]
Power	[ML²T⁻³]
Pressure	[ML⁻¹T⁻²]
Density	[ML⁻³]
Gravitational constant (G)	[M⁻¹L³T⁻²]
Universal gas constant (R)	[ML²T⁻²K⁻¹]
Planck’s constant (h)	[ML²T⁻¹]
Coulomb’s constant (k)	[M⁻¹L³T⁻⁴A²]
Charge	[AT]
Potential difference	[ML²T⁻³A⁻¹]
Resistance	[ML²T⁻³A⁻²]

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5. Uses of Dimensional Analysis

(A) To check correctness of an equation

Both sides must have the same dimensions (principle of homogeneity).

Example:

$$s=ut+\frac{1}{2}a{t}^2$$ All terms → [L] ✔

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(B) To derive relationships between physical quantities

Example: For time period of a pendulum:

$$T\propto \sqrt{\frac{L}{g}}$$

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(C) To convert units

Example:
1 N = 10⁵ dyne (derived using dimensions)

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6. Limitations of Dimensional Analysis (Frequently Asked Question)

• Cannot determine dimensionless constants (e.g., ½, 2π).

• Cannot differentiate between quantities with same dimensions (e.g., work & torque).

• Cannot be applied to equations involving trigonometric, logarithmic, exponential functions.

• Only checks correctness, not exact form.

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7. Important Dimensionless Numbers

Often asked in exams:

• Refractive index

• Relative density

• Strain

• Poisson’s ratio

• Coefficient of friction

• Reynolds number

All of the above → dimensionless

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8. Common Unit Conversions (High-Yield)

• 1 Å = 10⁻¹⁰ m

• 1 micron = 10⁻⁶ m

• 1 litre = 10⁻³ m³

• 1 eV = 1.6 × 10⁻¹⁹ J

• 1 bar = 10⁵ Pa

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9. Important Physical Constants (With Units)

• Speed of light, c = 3×10⁸ m/s

• Planck’s constant, h = 6.63×10⁻³⁴ Js

• Gravitational constant, G = 6.67×10⁻¹¹ Nm²/kg²

• Boltzmann constant, k = 1.38×10⁻²³ J/K

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10. Expected Exam Question Types

1. Match the following: quantities ↔ dimensions

2. Find dimensional formula of (h/e), (G/c²), etc.

3. Check correctness of an equation

4. Convert units (N to dyne, J to erg, Pa to bar)

5. Identify dimensionless quantities

6. Find dependence using dimensional analysis