Unit and Dimensions Theory Notes PDF for Class 11 Physics – Info Edu Story
Unit and Dimensions Theory Notes PDF Download – Class 11 Physics
For a deeper understanding of Chapter 1 UNITS AND DIMENSIONS Class 11 Physics Handwritten Notes, students can refer to our detailed Units-and-Measurements-Class-11-Notes-for-JEE. These Units and Measurements of Class 11 Notes provide clear explanations of all concepts, making them ideal for both school exams and competitive exams. You can also download the Unit and dimensions theory notes PDF for comprehensive study and easy revision anytime, anywhere.
Units and Measurements Notes | Class 11 Physics Notes
Physical
Quantity : -
The quantity
that can be measured and in which the laws of physics are expressed is called a
Physical Quantity. For example, length, time, mass, temperature, etc.
Physical
quantity (Q) = Magnitude × Unit = n × u
Where, n
represents the numerical value and u represents the unit.
Thus, while
expressing definite amount of physical quantity, it is clear that as the unit(u)
changes, the magnitude(n) will also change but product ‘nu’ will remain same.
i.e. n u =
constant, or n1 u1 = n2 u2 = constant
Fundamental
Physical Quantity: -
(1)
Fundamental Quantities:
Physical
quantities that are independent and need no other quantity for definition are
called fundamental or basic quantities. All other quantities are based on them.
There are Seven base units or fundamental quantities.
|
Quantity |
Unit |
Symbol |
|
1. Length |
metre |
m |
|
2. Mass |
kilogram |
kg |
|
3. Time |
second |
s |
|
4. Electric Current |
ampere |
A |
|
5. Temperature |
kelvin |
K |
|
6. Amount of Substance |
mole |
mol |
|
7. Luminous Intensity |
candela |
cd |
(2) Derived
Quantities:
Quantities
obtained by combining powers of fundamental quantities (by multiplication or
division) are derived quantities.
Example: Area = length², Volume = length³.
(3) Supplementary Physical Quantities: - Supplementary unit have no dimension but have units.
|
Quantity |
Unit |
Symbol |
|
Plane angle |
Radian |
rad |
|
Solid angle |
Steradian |
Sr |
Practical
Units
(1) Length:
1 fermi (fm) = 10⁻¹⁵ m
1 X-ray unit (XU) = 10⁻¹³ m
1 angstrom (Å) = 10⁻¹⁰ m = 10⁻⁸ cm = 0.1 μm
1 micron (μm) = 10⁻⁶ m
1 astronomical unit (A.U.) ≈ 1.5 × 10¹¹ m = 1.5 × 10⁸ km
1 light year (ly) = 9.46 × 10¹⁵ m
1 parsec (pc) = 3.26 light years
(2) Mass:
1 Chandra Shekhar unit (CSU) = 1.4 × mass of Sun = 2.8 × 10³⁰ kg
1 metric tonne = 1000 kg
1 quintal = 100 kg
1 atomic mass unit (amu) = 1.67 × 10⁻²⁷ kg (≈ mass of a proton/neutron)
(3) Time:
Year: Time taken by Earth to revolve once around the Sun.
Lunar month: Time taken by Moon to revolve once around Earth = 27.3 days.
Solar day: Time for one rotation of Earth with respect to the Sun; average solar day = mean of all days in a year.
1 solar year = 365.25 average solar days.
Sidereal day: Time for Earth’s one rotation with respect to a distant star.
1 solar year = 366.25 sidereal days = 365.25 solar days.
Hence, 1 sidereal day < 1 solar day.
Shake: 1 shake = 10⁻⁸ second (obsolete unit)
Dimension of Physical Quantity : -
When a derived quantity is written in terms of fundamental quantities, the powers of those quantities are called its dimensions.
Example: Force = mass × acceleration = M × L × T-2
So, Dimensions of Force: 1 in mass, 1 in length, –2 in time.
Dimensional Equation: [Force] = [M¹L¹T⁻²]
Dimensional Formula: [M L T⁻²]
Dimensional
Formula– It is an
expression of how the fundamental dimensions are combined to define a derived
quantity.
Dimensional
Equation- Dimensional
Equation is formed by equating a physical quantity with its dimensional
formula. Understanding the relationship between the fundamental dimensions that
form a physical quantity.
Principle of
Homogeneity- The
principle of homogeneity states that the dimensions of each term of a
dimensional equation on both sides must be the same. This principle helps to
convert the units from one form to another.
Quantities
Having Same Dimensions : -
1. [M⁰L⁰T⁻¹] → Frequency, Angular velocity, Decay
constant, Velocity gradient
2. [M¹L²T⁻²] → Work, Energy (K.E., P.E.), Torque,
Moment of force
3. [M¹L⁻¹T⁻²]
→ Pressure, Stress, Modulus (Young’s, Bulk, Rigidity), Energy density
4. [M¹L¹T⁻¹] → Momentum, Impulse
5. [M⁰L¹T⁻²] → Acceleration, g (Gravitational
field intensity)
6. [M¹L¹T⁻²] → Force, Weight, Thrust, Energy
gradient
7. [M¹L²T⁻¹] → Angular momentum, Planck’s constant
(h)
8. [M¹L⁰T⁻²] → Surface tension, Surface energy
9. [M⁰L⁰T⁰] →
Strain, Refractive index, Relative density, Angle, Solid angle, εr, μr
10. [M⁰L²T⁻²] → Latent heat, Gravitational
potential
11. [M¹L²T⁻²θ⁻¹]
→ Thermal capacity, Gas constant (R), Boltzmann constant (k), Entropy
12. [M⁰L⁰T¹] →
(√l/g), (m/k)¹ᐟ², (R/g)¹ᐟ², also L/R, LC, RC (in electricity)
13. [M¹L²T⁻²] → Energy forms in electricity:
(I²Rt), (VIt), (½LI²), (½CV²), etc.
Important
Dimensions in Heat
|
Quantity |
Unit |
Dimension |
|
Temperature (T) |
K |
[M⁰L⁰T⁰θ¹] |
|
Heat (Q) |
J |
[ML²T⁻²] |
|
Specific heat (c) |
J/kg·K |
[M⁰L²T⁻²θ⁻¹] |
|
Thermal capacity |
J/K |
[M¹L²T⁻²θ⁻¹] |
|
Latent heat (L) |
J/kg |
[M⁰L²T⁻²] |
|
Gas constant (R) |
J/mol·K |
[M¹L²T⁻²θ⁻¹] |
|
Boltzmann constant (k) |
J/K |
[M¹L²T⁻²θ⁻¹] |
|
Thermal conductivity (K) |
J/m·s·K |
[M¹L¹T⁻³θ⁻¹] |
|
Stefan’s constant (σ) |
W/m²·K⁴ |
[M¹L⁰T⁻³θ⁻⁴] |
|
Wien’s constant (b) |
m·K |
[M⁰L¹T⁰θ¹] |
|
Planck’s constant (h) |
J·s |
[M¹L²T⁻¹] |
|
Coefficient of linear expansion (α) |
K⁻¹ |
[M⁰L⁰T⁰θ⁻¹] |
|
Mechanical equivalent of heat (J) |
J/cal |
[M⁰L⁰T⁰] |
|
van der Waals constant (a) |
N·m⁴ |
[M¹L⁵T⁻²] |
|
van der Waals constant (b) |
m³ |
[M⁰L³T⁰] |
Applications
of Dimensional Analysis : -
(1) Finding
Unit of a Physical Quantity
Write the
formula, find dimensions, replace M, L, T by units of desired system.
Example: Work =
Force × Displacement
=> [W] =
[MLT⁻²][L] =
[ML²T⁻²]
Unit in CGS =
g·cm²/s² = erg
Unit in MKS =
kg·m²/s² = joule
(2) Finding
Dimensions of Constants or Coefficients
(i)
Gravitational Constant (G): F = G m1 m2 / r²
=> [G] =
[F][r²]/[m²] = [M⁻¹L³T⁻²]
(ii) Planck’s
Constant (h):
E = h ν
=> [h] =
[E]/[ν] = [ML²T⁻¹]
(iii)
Coefficient of Viscosity (η):
η = pr⁴ / (8l ×
dV/dt)
=> [η] = [ML⁻¹T⁻¹]
(3)
Converting from One System to Another
If [X] = [MᵃLᵇTᶜ],
then
n₂ = n₁ × (M₁/M₂)ᵃ
× (L₁/L₂)ᵇ × (T₁/T₂)ᶜ
Examples:
1 N = 10⁵ dyne
G (CGS → MKS):
6.67 × 10⁻⁸ → 6.67 ×
10⁻¹¹
(4) Checking
Dimensional Correctness
Principle of
Homogeneity: Dimensions of all terms on both sides of an equation must be same.
Example 1: F =
mv²/r → Not dimensionally correct.
Example 2: s =
ut + ½at² → Dimensionally correct.
(5) Deriving
Relations
(i) Time
Period of Simple Pendulum:
T ∝ mˣ lʸ gᶻ
[T] = [M⁰L⁰T¹]
= [MˣLʸ⁺ᶻT⁻²ᶻ]
=> x=0,
y=1/2, z=–1/2
=> T =
K√(l/g), experimentally K = 2π
Final: T =
2π√(l/g)
(ii) Stokes’
Law:
F ∝ ηˣ rʸ vᶻ
[F] = [MLT⁻²], [η] = [ML⁻¹T⁻¹]
=> x=y=z=1
=> F = K η r
v, experimentally K = 6π
Final: F = 6π η
r v
Limitations
of Dimensional Analysis
1. Not
Unique: Same
dimensional formula may represent many quantities.
Example: [M¹L²T⁻²] → work, energy, or torque.
2. Cannot
Find Numerical Constants:
Dimensionless
constants like ½, 2π, etc., cannot be determined by this method.
3. Fails for
Non-Power Relations:
Equations
involving trigonometric, exponential, or additive terms (like s = ut + ½at² or
y = a sin ωt) cannot be derived.
4. Limited
to ≤ 3 Variables:
If a quantity
depends on more than three variables, equations become insufficient.
Example: T =
2π√(l/g) cannot be derived, only checked.
5. Fails if
Two Variables Have Same Dimensions:
If two
variables have identical dimensions, the relation cannot be derived.
Example:
Frequency of tuning fork f = (1/2L)√(T/m) cannot be derived dimensionally.
Significant
Figures : -
Significant
figures indicate digits in which we have confidence.
More
significant figures → greater accuracy.
Rules for
Counting Significant Figures:
1. All non-zero
digits are significant
Examples: 42.3
→ 3 sf, 243.4 → 4 sf, 24.123 → 5 sf
2. Zeros
between non-zero digits are significant
Examples: 5.03
→ 3 sf, 5.604 → 4 sf, 4.004 → 4 sf
3. Leading
zeros (left of number) are NOT significant
Examples: 0.543
→ 3 sf, 0.045 → 2 sf, 0.006 → 1 sf
4. Trailing
zeros (right of number) ARE significant
Examples: 4.330
→ 4 sf, 433.00 → 5 sf, 343.000 → 6 sf
5. Exponential
notation: numerical part shows significant figures
Examples: 1.32
× 10⁻² → 3 sf,
1.32 × 10⁴ → 3 sf
Rounding Off
Rules
i. Digit
< 5: preceding digit unchanged
Examples: 7.82
→ 7.8, 94.3 → 94.3 → 94.3 (if rounding to 1 sf)
ii. Digit
> 5: preceding digit +1
Examples: 6.87
→ 6.9, 12.78 → 12.8
iii. Digit =
5 followed by non-zero digits: preceding digit +1
Examples:
16.351 → 16.4, 6.758 → 6.8
iv. Digit =
5 or 5 followed by zeros, preceding digit even: unchanged
Examples: 3.250
→ 3.2, 12.650 → 12.6
v. Digit = 5
or 5 followed by zeros, preceding digit odd: +1
Examples: 3.750
→ 3.8, 16.150 → 16.2
Errors of
Measurement : -
Definition:
Difference between measured value and true value of a quantity.
1. Absolute
Error (Δa) : -
·
Magnitude
of difference between true value and measured value.
·
If
measured values = a₁,
a₂, …, aₙ and mean = aₘ:
·
Δ₁ = a₁ – aₘ, Δ₂
= a₂ – aₘ, …, Δₙ = aₙ – aₘ
·
Absolute
errors can be positive or negative.
2. Mean
Absolute Error (Δā) : -
·
Arithmetic
mean of absolute errors:
·
Δā
= (|Δ₁| + |Δ₂| + … + |Δₙ|)/n
·
Final
measurement: aₘ ± Δā →
likely range: (aₘ
– Δā) to (aₘ + Δā)
3. Relative
/ Fractional Error : -
Fractional
error = Δā / aₘ
4.
Percentage Error : -
% Error = (Δā
/ aₘ) × 100
