NEET UG Physics · Chapter 1

Units and Measurements

SI units, dimensional analysis, significant figures, errors in measurement, vernier callipers and screw gauge.

NCERT Class 11NEET: 1–2 Qs per paper 13 min read

1. Chapter at a Glance

Physics is built on measurement. A measured value always has two parts — a number and a unit — and it is never perfectly exact. This NCERT Class 11 chapter explains the SI system of units, how to use dimensions to check formulas, how to report results with correct significant figures, and how errors combine when we calculate a quantity from measured values. In NEET, this chapter gives one or two easy-to-score questions every year, usually on dimensions or percentage error.

What NEET asks most
  • Dimensional formula of a physical quantity or constant.
  • Pairs of quantities having the same dimensions.
  • Maximum percentage error in a derived quantity.
  • Counting significant figures and rounding off.
  • Least count and reading of vernier callipers and screw gauge.

2. The SI System

The International System (SI) has seven base quantities, each with a base unit. Every other quantity — speed, force, energy, charge — is a derived quantity whose unit is made from these.

Base quantityUnit (symbol)DimensionNEET Note
Lengthmetre (m)[L]Defined using the speed of light
Masskilogram (kg)[M]Defined using Planck's constant
Timesecond (s)[T]Defined by caesium atomic clock
Electric currentampere (A)[A]Charge is a derived quantity (A s)
Temperaturekelvin (K)[K]Always use kelvin in gas laws
Amount of substancemole (mol)[mol]6.022 × 10²³ entities
Luminous intensitycandela (cd)[cd]Rarely asked

Two supplementary units are dimensionless: the radian (plane angle) and the steradian (solid angle). Some large and small practical units: 1 light-year ≈ 9.46 × 10¹⁵ m, 1 astronomical unit ≈ 1.5 × 10¹¹ m, 1 parsec ≈ 3.08 × 10¹⁶ m, 1 ångström = 10⁻¹⁰ m, 1 fermi = 10⁻¹⁵ m.

3. Dimensions and Their Uses

The dimensions of a quantity are the powers to which the base quantities must be raised to represent it. For example, velocity = length/time, so its dimensional formula is [M⁰LT⁻¹]; force = mass × acceleration = [MLT⁻²].

QuantityDimensionsSame dimensions as
Work, energy, torque[ML²T⁻²]Heat, moment of force
Power[ML²T⁻³]—
Momentum, impulse[MLT⁻¹]Each other
Pressure, stress, modulus[ML⁻¹T⁻²]Energy density
Angular momentum, Planck's constant[ML²T⁻¹]Each other
Surface tension, spring constant[MT⁻²]Force per length
Gravitational constant G[M⁻¹L³T⁻²]—
Coefficient of viscosity[ML⁻¹T⁻¹]—
Frequency, angular velocity[T⁻¹]Each other
Strain, angle, refractive indexdimensionless—

Three uses

  1. Checking an equation — the principle of homogeneity says every term added or equated must have the same dimensions.
  2. Converting units — for a quantity of dimensions [MaLbTc], n₂ = n₁(M₁/M₂)a(L₁/L₂)b(T₁/T₂)c.
  3. Deriving a relation — when a quantity depends on others as a product of powers.
Limitations

Dimensional analysis cannot give numerical constants (such as 2π or ½), cannot handle relations with sums of different terms, and fails for trigonometric, exponential and logarithmic functions. A dimensionally correct equation is not necessarily a correct equation.

4. Accuracy, Precision and Errors

  • Accuracy is closeness to the true value; precision is closeness of repeated readings to each other (decided by the least count).
  • Systematic errors push results in one direction (zero error, faulty calibration, wrong method). Random errors change unpredictably and are reduced by taking the mean of many readings.
  • Absolute error Δa = |amean − ai|; mean absolute error = average of these; relative error = Δa/a; percentage error = (Δa/a) × 100.
For Z = ApBq/Cr:   ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C)

For Z = A + B or Z = A − B, the absolute errors add: ΔZ = ΔA + ΔB. Errors always add, even when quantities are subtracted or divided.

Significant figures

  • All non-zero digits are significant. Zeros between non-zero digits are significant.
  • Leading zeros (as in 0.0045) are never significant. Trailing zeros after a decimal point (as in 4.500) are significant.
  • In multiplication/division, the answer keeps as many significant figures as the least precise factor. In addition/subtraction, it keeps as many decimal places as the least precise term.
  • Changing units does not change the number of significant figures: 2.50 m = 250 cm = 2.50 × 10² cm (three significant figures).

5. Vernier Callipers and Screw Gauge

anvil spindle wire sleeve (main scale) thimble (circular scale) LC = pitch ÷ number of circular divisions
Screw gauge: one full turn of the thimble moves the spindle by one pitch; each circular division moves it by the least count.
  • Vernier callipers: least count LC = 1 MSD − 1 VSD = (1 MSD)/(number of vernier divisions). Reading = MSR + (coinciding vernier division × LC).
  • Screw gauge: LC = pitch/number of circular divisions. Reading = MSR + (CSR × LC).
  • Zero error is positive when the zero of the moving scale lies ahead of the fixed zero, negative when behind. True reading = observed reading − zero error (with its sign).

6. Formula Master Table

SituationFormula
Unit conversionn₂ = n₁(M₁/M₂)a(L₁/L₂)b(T₁/T₂)c
Product or powerΔZ/Z = pΔA/A + qΔB/B + rΔC/C
Sum or differenceΔZ = ΔA + ΔB
Percentage error(Δa/a) × 100
Vernier least countLC = 1 MSD − 1 VSD
Screw gauge least countLC = pitch/N
True readingobserved − zero error

7. Master Summary Table

TopicKey ideaNEET Note
SI units7 base + 2 dimensionless supplementary unitsRadian and steradian have no dimensions
DimensionsPowers of M, L, T, A, KLearn 10 standard formulas; derive the rest
HomogeneityOnly like quantities addArguments of sin, log, ex must be dimensionless
ErrorsRelative errors add with powersQuantity with highest power matters most
Significant figuresLeading zeros never countRound the final answer only
InstrumentsLC decides precisionSubtract zero error with sign

8. Mind Map

Units & Measurements
Units
  • 7 SI base units
  • Derived units
  • Practical units
Dimensions
  • Check equations
  • Convert units
  • Derive relations
Errors
  • Systematic, random
  • Propagation
  • Significant figures
Instruments
  • Vernier callipers
  • Screw gauge
  • Zero error

9. Tips and Tricks

Shortcuts
  • To find the dimensions of a constant, rewrite the formula it appears in: G = Fr²/m₁m₂, h = E/ν, η = F/(A dv/dx).
  • Energy/volume has the dimensions of pressure — useful in elasticity and electrostatics questions.
  • In error questions, multiply each percentage error by the absolute value of its power; square roots contribute half.
  • For "which is dimensionless" questions, look for ratios of like quantities (strain, relative density, μ₀ε₀c²).
Common NEET traps
  • Subtracting percentage errors when one quantity divides another — they always add.
  • Counting the zeros in 0.0050 as significant — only "50" counts (2 significant figures).
  • Confusing a unit being "dimensionless" with it having no unit (radian has a unit but no dimension).

10. Important One-liners

  • Torque and work have the same dimensions but torque is a vector and work a scalar.
  • Impulse and linear momentum have identical dimensions [MLT⁻¹].
  • Planck's constant and angular momentum both have dimensions [ML²T⁻¹].
  • Light-year is a unit of distance, not of time.
  • Precision of an instrument depends on its least count.
  • Mean of many readings reduces random error but not systematic error.
  • The number of significant figures does not depend on the choice of units.

11. Solved Numericals: Average to Advanced

Five average and five advanced numericals. Try each yourself, then click the button to see the answer with step-by-step working.

Average level

N1Average

In a new system, the unit of mass is 5 kg, the unit of length is 2 m and the unit of time is 1 s. Express an energy of 50 J in this system.

Show answer & solution

Given: n₁ = 50, M₂ = 5 kg, L₂ = 2 m, T₂ = 1 s. Concept: energy has dimensions [ML²T⁻²].

  1. n₂ = n₁(M₁/M₂)¹(L₁/L₂)²(T₁/T₂)⁻².
  2. = 50 × (1/5) × (1/2)² × 1.
  3. = 50 × 0.2 × 0.25 = 2.5.

Answer: 2.5 new units of energy

N2Average

The radius of a sphere is measured with an error of 0.5%. What is the percentage error in its volume?

Show answer & solution

Concept: V = (4/3)πr³; constants have no error.

  1. ΔV/V = 3(Δr/r).
  2. = 3 × 0.5% = 1.5%.

Answer: 1.5%

N3Average

The length and breadth of a rectangle are 12.5 cm and 3.2 cm. Find its area with the correct number of significant figures.

Show answer & solution

Concept: in multiplication, keep the significant figures of the least precise factor.

  1. Raw product = 12.5 × 3.2 = 40.00 cm².
  2. 3.2 has two significant figures, so the answer must have two.
  3. Area = 40 cm² (written as 4.0 × 10¹ cm² to show two significant figures).

Answer: 4.0 × 10¹ cm²

N4Average

A vernier calliper has 20 vernier divisions equal to 19 main-scale divisions of 1 mm each. The main-scale reading is 32 mm and the 8th vernier division coincides. Find the least count and the reading (no zero error).

Show answer & solution

Concept: LC = 1 MSD − 1 VSD.

  1. 1 VSD = 19/20 mm = 0.95 mm ⇒ LC = 1 − 0.95 = 0.05 mm.
  2. Vernier part = 8 × 0.05 = 0.40 mm.
  3. Reading = 32 + 0.40 = 32.40 mm.

Answer: 0.05 mm; 32.40 mm

N5Average

The mass of a body is measured with a 1% error and its speed with a 2% error. What is the maximum percentage error in its kinetic energy?

Show answer & solution

Concept: K = ½mv²; the factor ½ has no error.

  1. ΔK/K = Δm/m + 2(Δv/v).
  2. = 1% + 2 × 2% = 5%.

Answer: 5%

Advanced level

N6Advanced

The speed v of a transverse wave on a stretched string depends on the tension T and the mass per unit length μ. Use dimensions to find the relation.

Show answer & solution

Concept: let v = kTaμb; T is a force [MLT⁻²], μ is [ML⁻¹].

  1. [LT⁻¹] = [MLT⁻²]a[ML⁻¹]b = Ma+b La−b T−2a.
  2. T: −2a = −1 ⇒ a = ½.
  3. M: a + b = 0 ⇒ b = −½. (Check L: a − b = 1 ✓.)
  4. So v = k√(T/μ); experiment gives k = 1.

Answer: v = √(T/μ)

N7Advanced

A screw gauge has a pitch of 1 mm and 100 circular divisions. With the studs closed, the zero of the circular scale is 4 divisions below the reference line (positive zero error). For a wire, the main scale reads 2 mm and the circular scale reads 56. Find the true diameter.

Show answer & solution

Concept: LC = pitch/N; true reading = observed − zero error.

  1. LC = 1/100 = 0.01 mm.
  2. Zero error = +4 × 0.01 = +0.04 mm.
  3. Observed = 2 + 56 × 0.01 = 2.56 mm.
  4. True diameter = 2.56 − 0.04 = 2.52 mm.

Answer: 2.52 mm

N8Advanced

A resistance is found from R = V/I with V = (50 ± 1) V and I = (5.0 ± 0.2) A. Find R with its absolute error.

Show answer & solution

Concept: for a quotient, percentage errors add.

  1. R = 50/5.0 = 10 Ω.
  2. ΔV/V = 1/50 = 2%; ΔI/I = 0.2/5.0 = 4%.
  3. ΔR/R = 2% + 4% = 6%.
  4. ΔR = 6% of 10 = 0.6 Ω.

Answer: R = (10 ± 0.6) Ω

N9Advanced

A pendulum of length (50.0 ± 0.1) cm makes 20 oscillations in (28.0 ± 0.2) s. Find g and its percentage error.

Show answer & solution

Concept: g = 4π²L/T²; the error in T equals the error in total time divided by 20, so ΔT/T = Δt/t.

  1. T = 28.0/20 = 1.40 s.
  2. g = 4 × 9.87 × 0.500/1.96 ≈ 10.1 m s⁻².
  3. ΔL/L = 0.1/50.0 = 0.2%; ΔT/T = 0.2/28.0 ≈ 0.71%.
  4. Δg/g = 0.2% + 2 × 0.71% ≈ 1.6%.

Answer: g ≈ 10.1 m s⁻² with about 1.6% error

N10Advanced

The temperatures of two bodies are (60.0 ± 0.5) °C and (40.0 ± 0.3) °C. Find the temperature difference with its absolute and percentage error.

Show answer & solution

Concept: in a difference, absolute errors add — this is why subtracting close values gives large percentage errors.

  1. ΔTvalue = 60.0 − 40.0 = 20.0 °C.
  2. Absolute error = 0.5 + 0.3 = 0.8 °C.
  3. Percentage error = (0.8/20.0) × 100 = 4%.

Answer: (20.0 ± 0.8) °C, i.e. 4%

12. NEET-pattern Questions with Detailed Answers

These questions are original and modelled on concepts repeatedly asked in NEET UG.

Q1Easy

The dimensional formula of pressure is:

  1. [MLT⁻²]
  2. [ML⁻¹T⁻²]
  3. [ML²T⁻²]
  4. [ML⁻²T⁻¹]
Show solution

Pressure = force/area = [MLT⁻²]/[L²] = [ML⁻¹T⁻²]. (A) is force, (C) is energy, (D) matches nothing standard.

Answer: (B)

Q2Easy

Which pair has the same dimensions?

  1. Impulse and momentum
  2. Force and pressure
  3. Work and power
  4. Stress and strain
Show solution

Impulse = F × t = [MLT⁻¹] = momentum. Force and pressure differ by area, work and power differ by time, and strain is dimensionless while stress is not.

Answer: (A)

Q3Easy

The number of significant figures in 0.02030 is:

  1. 2
  2. 3
  3. 4
  4. 5
Show solution

The leading "0.0" is not significant. The digits 2, 0 (between non-zeros), 3 and the trailing 0 after the decimal point are significant: 4.

Answer: (C)

Q4Medium

The SI unit of the universal gravitational constant G is:

  1. N m kg⁻²
  2. N m² kg⁻²
  3. N kg² m⁻²
  4. N m² kg²
Show solution

From F = Gm₁m₂/r², G = Fr²/(m₁m₂), so the unit is N m² kg⁻².

Answer: (B)

Q5Medium

If the speed of a body is measured with a 3% error (mass known exactly), the error in its kinetic energy is:

  1. 1.5%
  2. 3%
  3. 6%
  4. 9%
Show solution

K ∝ v², so ΔK/K = 2 × 3% = 6%. Option (D) would apply for a cube.

Answer: (C)

Q6Medium

A screw gauge has a pitch of 0.5 mm and 50 divisions on its circular scale. Its least count is:

  1. 0.1 mm
  2. 0.01 mm
  3. 0.001 mm
  4. 0.05 mm
Show solution

LC = pitch/N = 0.5/50 = 0.01 mm.

Answer: (B)

Q7Hard

In the relation y = A sin(ωt − kx), where x is distance and t is time, the dimensions of ω/k are those of:

  1. Acceleration
  2. Velocity
  3. Frequency
  4. Wavelength
Show solution

The argument of sine must be dimensionless: ω = [T⁻¹] and k = [L⁻¹]. So ω/k = [LT⁻¹], a velocity (it is the wave speed).

Answer: (B)

Q8Hard

A physical quantity is P = a²b³/(c√d). The percentage errors in a, b, c and d are 1%, 2%, 3% and 4%. The maximum percentage error in P is:

  1. 10%
  2. 11%
  3. 13%
  4. 15%
Show solution

ΔP/P = 2(1%) + 3(2%) + 1(3%) + ½(4%) = 2 + 6 + 3 + 2 = 13%. Errors add even for quantities in the denominator.

Answer: (C)

13. Quick Revision Summary

Remember before the exam
  • Seven SI base units; radian and steradian are dimensionless.
  • Only like dimensions can be added; arguments of sin, log and ex are dimensionless.
  • Percentage errors add with their powers; absolute errors add in sums and differences.
  • Leading zeros are never significant; round only the final answer.
  • Vernier LC = 1 MSD − 1 VSD; screw gauge LC = pitch/N; true = observed − zero error.
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