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JEE Main (Engineering) · Physics (JEE & NEET)

Units and Measurements

SI units, dimensions, significant figures, error analysis and least-count based instrument readings used across JEE Physics.

Six measurement ideas that recur inside almost every numerical JEE Physics question: units, dimensions, significant figures, least count and error propagation.

  • JEE Main (Engineering)
  • Medium level
  • 5 concepts
  • 5 practice questions

1SI units and dimensions

A unit tells you how much of a quantity you have; a dimension tells you what kind of quantity it is. Metre and centimetre are different units of the same dimension [L], while metre and second cannot be added because their dimensions differ. JEE questions often hide this in derived units: a newton is kg\,m\,s^{-2}, so force is [MLT^{-2}].

Figure. A unit is a size; a dimension is a kind. Newton, joule and watt are different packages of M, L and T — equal bar heights here only mark that each is one coherent SI derived unit, not that the quantities are equal.

How it works

  1. Reduce the unitWrite every named unit in base SI units before comparing it.
  2. Read powersThe powers of M, L and T give the dimension of the quantity.
  3. Compare like with likeOnly terms with identical dimensions can be added or equated.
Derived units worth reducing once
QuantityUnitDimension
ForceN = kg m s^-2[MLT^{-2}]
EnergyJ = N m[ML^2T^{-2}]
PressurePa = N m^-2[ML^{-1}T^{-2}]
PowerW = J s^-1[ML^2T^{-3}]
The dimensions of pressure are
  1. [MLT^{-2}]
  2. [ML^{-1}T^{-2}]
  3. [ML^2T^{-2}]

Pressure is force per area, so [MLT^{-2}]/[L^2]=[ML^{-1}T^{-2}].

2Dimensional analysis and its limits

Dimensional analysis can check whether an equation is possible and can predict powers in a proportional law. If the period of a simple pendulum depends only on length l and acceleration g, write T\propto l^a g^b; matching dimensions gives a=1/2 and b=-1/2. The method cannot discover the missing factor 2\pi.

Figure. Dimensional analysis accepts √(l/g) for a pendulum period and rejects a guess with the wrong powers. It cannot fix dimensionless prefactors like 2π — that needs dynamics.

How it works

  1. Assume powersWrite the target quantity as a product of possible variables raised to unknown powers.
  2. Match dimensionsEquate powers of M, L and T on both sides.
  3. Remember constantsPure numbers and functions of angles are invisible to dimensional analysis.

Pendulum period up to a constant

Use dimensions to find how the period T depends on length l and g.

  • T\propto l^a g^b[T]=[L]^a[LT^{-2}]^b
  • Powers of T: -2b=1b=-1/2
  • Powers of L: a+b=0a=1/2
  • DependenceT\propto\sqrt{l/g}

Pro tip. The real formula is 2\pi\sqrt{l/g}; dimensions got the powers, not the number.

Dimensional analysis can determine
  1. The factor 2\pi in a pendulum period
  2. Whether two terms may be added
  3. The numerical value of coefficient of friction

It checks dimensions and powers. Dimensionless constants and dimensionless coefficients are invisible to it.

3Significant figures are a precision contract

A recorded number carries the precision of the measurement. In 0.00420\text{ m}, the leading zeros only place the decimal point and are not significant; the final zero after 2 is significant because it says the instrument resolved that place. In products and quotients, the answer keeps the fewest significant figures among measured inputs.

Figure. Leading zeros place the decimal; trailing zeros after a decimal are measured. Scientific notation removes the ambiguity that bare trailing zeros in 4200 create.

How it works

  1. Ignore leading zerosThey locate the decimal point but do not count as measured digits.
  2. Keep captive zerosZeros between non-zero digits, and trailing zeros after a decimal point, are significant.
  3. Round at the endDo not round intermediate rows in a multi-step calculation.
The number 0.00420 has how many significant figures?
  1. 2
  2. 3
  3. 5

The leading zeros are not significant. The digits 4, 2 and the final decimal zero are significant, so there are 3.

4Least count and instrument reading

Least count is the smallest change an instrument can directly resolve. A vernier reading is main-scale reading plus coinciding vernier division times least count, corrected for zero error. A screw gauge reading is pitch-scale reading plus circular-scale division times least count, again corrected for zero error.

Figure. Least count is the smallest step the instrument prints. Vernier and screw gauge buy finer LC at the cost of a zero-error correction on every reading.

How it works

  1. Main readingTake the last main-scale mark just before the vernier zero, or the pitch-scale reading on a screw gauge.
  2. CoincidenceMultiply the coinciding vernier or circular division by least count.
  3. Zero correctionSubtract positive zero error and add negative zero error.

A vernier reading

A caliper has least count 0.01\text{ cm}, main-scale reading 2.30\text{ cm} and 6th vernier coincidence. Find the length with zero error ignored.

  • Vernier contribution6\times0.01=0.06\text{ cm}
  • Total reading2.30+0.06=2.36\text{ cm}

Pro tip. The coinciding division is a multiplier of least count, not a centimetre reading.

A screw gauge has pitch 1\text{ mm} and 100 circular divisions. Its least count is
  1. 0.1\text{ mm}
  2. 0.01\text{ mm}
  3. 1\text{ mm}

Least count = pitch / circular divisions = 1 mm / 100 = 0.01 mm.

5Error propagation

Errors combine according to the operation. For addition or subtraction, add absolute errors because the measured quantities are being laid on the same scale. For multiplication, division and powers, add fractional errors; a power multiplies the fractional error by that power. For \rho=m/(4\pi r^3/3), the percentage error in r is tripled.

Figure. Sums add absolute errors; products and quotients add relative errors; a power multiplies the relative error by |n|. Match the rule to the operation in the formula.

How it works

  1. Sums use absolute errorsx+y and x-y both carry \Delta x+\Delta y as the maximum absolute error.
  2. Products use fractional errorsTurn each error into \Delta x/x before adding.
  3. Powers multiply errorsx^n contributes n\Delta x/x.

Density of a sphere

Mass has 1\% error and radius has 2\% error. Find the percentage error in density.

  • \rho=m/(4\pi r^3/3)\Delta\rho/\rho=\Delta m/m+3\Delta r/r
  • Percentage error1+3\times2=7\%

Pro tip. Constants such as 4/3 and \pi carry no measurement error.

If z=x^2/y and both x and y have 1\% error, the maximum percentage error in z is
  1. 2\%
  2. 3\%
  3. 1\%

x^2 contributes 2\% and division by y contributes another 1\%, so the maximum is 3\%.

Notes

  • SI units fix seven base quantities: length, mass, time, electric current, temperature, amount of substance and luminous intensity. Derived units such as newton, joule, pascal and volt are built from them; always reduce unfamiliar units to base dimensions before comparing expressions.
  • Dimensions track powers of base quantities, e.g. force has [MLT^{-2}], energy has [ML^2T^{-2}] and pressure has [ML^{-1}T^{-2}]. Dimensional homogeneity is necessary for any equation, so unlike dimensions cannot be added.
  • Dimensional analysis checks formulae, converts units and predicts powers in proportional laws, but it cannot determine dimensionless constants such as 2, \pi, coefficients of friction, angles in radians or refractive index.
  • A measured value must be reported with the instrument least count and the right significant figures. Zeros between non-zero digits are significant; leading zeros are not; trailing zeros after a decimal point are significant.
  • Errors may be absolute, relative or percentage. For products and quotients, fractional errors add; for powers, the fractional error is multiplied by the power. For sums and differences, absolute errors add.

Formulas

  • Relative error: \dfrac{\Delta x}{x}; percentage error: \dfrac{\Delta x}{x}\times100\%
  • Sum/difference: if z=x\pm y, then \Delta z=\Delta x+\Delta y
  • Product/quotient: if z=\dfrac{x^a y^b}{w^c}, then \dfrac{\Delta z}{z}=a\dfrac{\Delta x}{x}+b\dfrac{\Delta y}{y}+c\dfrac{\Delta w}{w}
  • Vernier least count: LC=1\,\text{MSD}-1\,\text{VSD}=\dfrac{1\,\text{MSD}}{n} for n vernier divisions spanning n-1 main-scale divisions
  • Screw gauge least count: LC=\dfrac{\text{pitch}}{\text{number of circular-scale divisions}}
  • Dimensional forms: [F]=[MLT^{-2}],\ [E]=[ML^2T^{-2}],\ [P]=[ML^{-1}T^{-2}]

Exam traps & shortcuts

  • If an answer option has the wrong dimension, eliminate it before substituting numbers; this catches many coefficient-heavy traps.
  • For multiplication by powers, multiply percentage error by the power: r^2 doubles the fractional error in r, and r^3 triples it.
  • Final numerical answers should not carry more significant figures than the least precise measured input unless the problem asks for an exact symbolic value.

Reference tables

Use the operation, not the look of the formula.

Measurement error rules
OperationError to addExample
x+y or x-yabsolute errors\Delta z=\Delta x+\Delta y
xy or x/yfractional errors\Delta z/z=\Delta x/x+\Delta y/y
x^nn times fractional error\Delta z/z=n\Delta x/x

Recap

Read only this before a measurement-heavy problem.

Units
Reduce named units to base SI units before comparing options.
Dimensions
Homogeneity is necessary, not sufficient; constants and trig functions are invisible.
Precision
Least count fixes the reading place, and significant figures record measured precision.
Errors
Sums use absolute error; products, quotients and powers use fractional error.

Practise Units and Measurements

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