01
CONCEPT
Units & Dimensions
In one line
Every physical quantity is number × unit. Its dimensions tell how it is built from M, L, T, A, K, mol and cd.
Every physical quantity is number × unit. Its dimensions tell how it is built from M, L, T, A, K, mol and cd.
PHYSICAL QUANTITY
1.1.1 Physical quantity
- n = numerical value, u = unit. A bigger unit gives a smaller number: n ∝ 1u.
- Fundamental quantities do not depend on others; derived quantities are built from them.
- Ratio quantities (relative density, refractive index, strain) have no unit.
SI UNITS
1.1.2 SI base units
| Quantity | 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 |
- Supplementary: radian (plane angle) and steradian (solid angle). Angles have units but no dimensions.
- CGS: cm, g, s. MKS: m, kg, s. FPS: foot, pound, second.
PREFIXES
1.1.3 Prefixes
| Power | Prefix | Power | Prefix |
|---|---|---|---|
| 10¹⁸ | exa (E) | 10⁻¹ | deci (d) |
| 10¹⁵ | peta (P) | 10⁻² | centi (c) |
| 10¹² | tera (T) | 10⁻³ | milli (m) |
| 10⁹ | giga (G) | 10⁻⁶ | micro (μ) |
| 10⁶ | mega (M) | 10⁻⁹ | nano (n) |
| 10³ | kilo (k) | 10⁻¹² | pico (p) |
| 10² | hecto (h) | 10⁻¹⁵ | femto (f) |
PRACTICAL UNITS
1.1.4 Practical units
| Unit | Value |
|---|---|
| 1 fermi | 10⁻¹⁵ m |
| 1 Å | 10⁻¹⁰ m |
| 1 AU | 1.5 × 10¹¹ m |
| 1 light year | 9.46 × 10¹⁵ m |
| 1 parsec | 3.26 ly = 3.08 × 10¹⁶ m |
| 1 amu | 1.66 × 10⁻²⁷ kg |
| 1 eV | 1.6 × 10⁻¹⁹ J |
| 1 kWh | 3.6 × 10⁶ J |
| 1 atm | 1.013 × 10⁵ N m⁻² |
- 1′ (minute of arc) = π180 × 60 rad ≈ 2.91 × 10⁻⁴ rad.
- Light year and parsec measure distance, not time.
DIMENSIONS
1.1.5 Dimensions and dimensional formula
- Dimensions = the powers to which M, L, T (and A, K) are raised to build a quantity.
- Force = mass × acceleration ⇒ [F] = [MLT⁻²].
- Pure numbers, angles, strain, refractive index: dimensionless [M⁰L⁰T⁰].
TABLE
1.1.6 Dimensions of common quantities
| Quantity | Dimensions |
|---|---|
| velocity | [LT⁻¹] |
| acceleration, g | [LT⁻²] |
| force, weight | [MLT⁻²] |
| momentum, impulse | [MLT⁻¹] |
| work, energy, torque | [ML²T⁻²] |
| power | [ML²T⁻³] |
| pressure, stress, Y | [ML⁻¹T⁻²] |
| surface tension, spring constant | [MT⁻²] |
| viscosity η | [ML⁻¹T⁻¹] |
| angular momentum, h | [ML²T⁻¹] |
| moment of inertia | [ML²] |
| G | [M⁻¹L³T⁻²] |
| latent heat | [L²T⁻²] |
| specific heat | [L²T⁻²K⁻¹] |
| gas constant R, k | [ML²T⁻²K⁻¹] |
| Stefan σ | [MT⁻³K⁻⁴] |
| Wien b | [LK] |
| frequency, ω, velocity gradient | [T⁻¹] |
SAME DIMENSIONS
1.1.7 Quantities with the same dimensions
- Work, energy, torque: [ML²T⁻²].
- Pressure, stress, Young’s modulus, energy density: [ML⁻¹T⁻²].
- Planck’s constant and angular momentum: [ML²T⁻¹].
- Impulse and momentum: [MLT⁻¹].
- Frequency, angular velocity, velocity gradient, decay constant: [T⁻¹]. √lg, √mk: [T].
JEE · NEET
Corner
Electrical and magnetic quantities
| Quantity | Dimensions |
|---|---|
| charge q | [AT] |
| potential V | [ML²T⁻³A⁻¹] |
| resistance R | [ML²T⁻³A⁻²] |
| capacitance C | [M⁻¹L⁻²T⁴A²] |
| inductance L | [ML²T⁻²A⁻²] |
| magnetic field B | [MT⁻²A⁻¹] |
| magnetic flux φ | [ML²T⁻²A⁻¹] |
| ε₀ | [M⁻¹L⁻³T⁴A²] |
| μ₀ | [MLT⁻²A⁻²] |
- Time: RC, LR, √(LC). Speed: 1√μ₀ε₀ = c.
- EB = velocity; μ₀ε₀ = R²; B²2μ₀ and 12ε₀E² = energy density.
02
CONCEPT
Applications of Dimensional Analysis
In one line
Every term of a correct equation has the same dimensions. Use it to find constants, convert units and guess formulas.
Every term of a correct equation has the same dimensions. Use it to find constants, convert units and guess formulas.
HOMOGENEITY
1.2.1 Principle of homogeneity
- Only like quantities can be added or subtracted.
- So every term on both sides of an equation has the same dimensions.
- A dimensionally correct equation may still be wrong (a missing ½ or 2π cannot be seen).
CONSTANTS
1.2.2 Dimensions of constants in an equation
- v = at + bt + c: c adds to t ⇒ [c] = [T]; at is a velocity ⇒ [a] = [LT⁻²]; [b] = [L].
- Van der Waals: (P + aV²)(V − b) = RT ⇒ [a] = [PV²] = [ML⁵T⁻²], [b] = [L³].
- The argument of sin, cos, log and ex is dimensionless.
CONVERSION
1.2.3 Conversion of units
- For a quantity of dimensions [MaLbTc]; system 1 is known, system 2 is new.
- 1 N = 10⁵ dyne; 1 J = 10⁷ erg.
- G = 6.67 × 10⁻⁸ CGS = 6.67 × 10⁻¹¹ SI.
DERIVING
1.2.4 Deriving a relation
- Assume the quantity ∝ product of powers of the quantities it depends on.
- Put in dimensions and equate the powers of M, L and T.
- Stokes’ law the same way: F = k ηx ry vz ⇒ x = y = z = 1, F = 6πηrv.
NEW BASE QUANTITIES
1.2.5 Other quantities as base
- If F, V, T (or any three) are chosen as base: write each of M, L, T in terms of them, then substitute.
- Mass = [FV⁻¹T], length = [VT] with F, V, T as base.
LIMITATIONS
1.2.6 Limitations
- Cannot find dimensionless constants (½, 2π).
- Cannot derive sums, or formulas with sin, log or ex.
- Fails if the quantity depends on more than three others.
- Same dimensions do not fix the quantity: [ML²T⁻²] may be work or torque.
JEE · NEET
Corner
Constants in exponents and trig functions
- In e−αt, sin(kx − ωt), log(x) the argument must be dimensionless.
- So [α] = [T⁻¹], [k] = [L⁻¹], [ω] = [T⁻¹].
- In F = a e−bx: [a] = [F] and [b] = [L⁻¹].
New base quantities
- Write the asked quantity as FaVbTc (or with the given base set).
- Equate the powers of M, L, T and solve.
- With F, V, T as base: mass = [FV⁻¹T], length = [VT].
- Planck length = √hGc³, Planck time = √hGc⁵.
03
CONCEPT
Significant Figures & Order of Magnitude
In one line
Significant figures are the digits we are sure of, plus one. The answer can never be more precise than the worst reading.
Significant figures are the digits we are sure of, plus one. The answer can never be more precise than the worst reading.
COUNTING
1.3.1 Counting significant figures
| Rule | Example |
|---|---|
| all non-zero digits count | 42.3 → 3 |
| zeros between non-zero digits count | 5.004 → 4 |
| leading zeros never count | 0.0045 → 2 |
| trailing zeros after a decimal count | 4.330 → 4 |
| in a × 10ⁿ only a counts | 1.32 × 10⁴ → 3 |
- Changing units does not change the count: 2.30 m = 230 cm = 2.30 × 10² cm (3 figures).
CALCULATIONS
1.3.2 Significant figures in calculations
- Add or subtract: keep the fewest decimal places.
- Multiply or divide: keep the fewest significant figures.
- Exact counts (7 squares, 2 in 2πr) do not limit the answer.
ROUNDING
1.3.3 Rounding off
- Digit dropped < 5: leave the last digit (7.82 → 7.8).
- Digit dropped > 5: raise it by one (6.87 → 6.9).
- Exactly 5: raise an odd digit, keep an even one (3.750 → 3.8, 3.250 → 3.2).
ORDER OF MAGNITUDE
1.3.4 Order of magnitude
- a ≤ 5: order = b. a > 5: order = b + 1.
- Earth’s rotation ≈ 10⁵ s, a year ≈ 10⁷ s, light wave period ≈ 10⁻¹⁵ s, sound wave period ≈ 10⁻³ s.
04
CONCEPT
Errors in Measurement
In one line
Errors always add. For a product or power, add the percentage errors, each multiplied by its power: the most asked MHT CET idea of this chapter.
Errors always add. For a product or power, add the percentage errors, each multiplied by its power: the most asked MHT CET idea of this chapter.
TYPES
1.4.1 Types of errors
| Error | Cause | Cure |
|---|---|---|
| systematic | faulty instrument, zero error, bad method | correct the setup |
| random | unpredictable changes (temperature, supply voltage) | repeat and average |
| least count | limited resolution of the scale | finer instrument |
| personal (gross) | carelessness, wrong reading | take care |
- Accurate = close to the true value. Precise = readings close to each other (small least count).
ABSOLUTE ERROR
1.4.2 Absolute, mean and relative error
- Result: a = am ± Δam.
- Absolute error has the unit of the quantity; relative error has no unit.
- A single reading: the error is its least count.
SUM AND DIFFERENCE
1.4.3 Error in a sum or difference
- Absolute errors add, even for a difference.
- % error = Δa + Δba ± b × 100: it is larger for a difference.
- Resistors in parallel: ΔRpRp² = ΔR₁R₁² + ΔR₂R₂².
PRODUCT AND POWER
1.4.4 Error in a product, quotient or power
- Powers multiply the error; a power in the denominator still adds.
- A square root halves the error: x = √a ⇒ ½ of the % error of a.
- The quantity with the highest power must be measured most carefully.
| Quantity | % error |
|---|---|
| area of a square (L²) | 2 × %L |
| volume of a sphere (r³) | 3 × %r |
| density of a cube (mL³) | %m + 3 %L |
| density of a wire (mπr²l) | %m + 2 %r + %l |
| kinetic energy (12mv²) | %m + 2 %v |
| pressure (FL²) | %F + 2 %L |
| g = 4π²LT² | %L + 2 %T |
- Pendulum: time n oscillations; ΔT ÷ T = least count of the watchtotal time.
- If V has 6% error, r has 2%, so the surface area has 4%.
05
CONCEPT
Least Count, Vernier Callipers & Screw Gauge
In one line
Least count is the smallest reading an instrument can take. Smaller least count ⇒ more precise reading.
Least count is the smallest reading an instrument can take. Smaller least count ⇒ more precise reading.
LEAST COUNT
1.5.1 Least count
| Instrument | Least count |
|---|---|
| metre scale | 1 mm |
| vernier callipers | 0.1 mm = 0.01 cm |
| screw gauge | 0.01 mm = 0.001 cm |
| stop watch | 0.1 s or 0.01 s |
| spherometer | 0.01 mm |
- A reading is written to the least count: 3.50 cm means LC 0.01 cm (vernier).
- The error of a single reading = least count.
- The last digit of a set of readings shows the least count.
JEE · NEET
Corner
Vernier callipers
Reading = 23 mm + 4 × 0.1 mm = 23.4 mm
- Reading = MSR + (coinciding vernier division) × LC.
- If m VSD = p MSD, then LC = (1 − p ÷ m) MSD.
- Zero error: vernier zero to the right of the main zero ⇒ positive; to the left ⇒ negative.
- Corrected reading = observed reading − zero error (with its sign).
- Negative zero error with the nth division coinciding = −(N − n) × LC, N = total vernier divisions.
Screw gauge
Object held between the anvil and the spindle
- Reading = MSR + (circular scale reading) × LC.
- Circular zero below the reference line ⇒ positive zero error; above it ⇒ negative.
- Corrected reading = observed − zero error.
Travelling microscope and spectrometer
- Both carry a vernier: VC = 1 MSD − 1 VSD.
- Spectrometer reads angles: MSD in degrees or minutes, so LC comes in minutes of arc.
EXAM-DAY CARD
Just see this before the exam
n₁u₁ = n₂u₂
units
[F] = [MLT⁻²]
dimensions
n₂ = n₁[M₁ ÷ M₂]a…
conversion
fewest decimals / figures
significant figures
Δxx = nΔaa + mΔbb
errors
LC = 1 MSD − 1 VSD
least count
FORMULA SHEET
Whole chapter on one page
Units
Q = nu, n ∝ 1u
7 SI base units + rad, sr
1 ly = 9.46 × 10¹⁵ m, 1 pc = 3.26 ly
Dimensions
energy, torque: [ML²T⁻²]
pressure, Y: [ML⁻¹T⁻²]
h, L: [ML²T⁻¹]; G: [M⁻¹L³T⁻²]
Dimensional analysis
every term: same dimensions
ex, sin x: x dimensionless
T = 2π√lg by dimensions
Significant figures
±: fewest decimal places
× ÷: fewest significant figures
order: a ≤ 5 ⇒ b, else b + 1
Errors
Δ(a ± b) = Δa + Δb
Δxx = nΔaa + mΔbb + pΔcc
g: %L + 2 %T
Instruments
vernier: LC = 1 MSD − 1 VSD
screw: LC = pitchN
true = reading − zero error
TOP MISTAKES
Avoid these = easy marks
MISTAKE ✗
In a − b the errors subtract
Error in r³ = error in r
Leading zeros are significant
Angle has no unit
CORRECT ✓
Absolute errors always add
Multiply by the power: 3 × %r
0.0045 has 2 significant figures
Angle has a unit (rad) but no dimension