Units and Measurements — Easy Practice (Set 1)

Quick Recap — Units and Measurements

A fast refresher before you practise. This set checks these ideas directly.

The 7 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

Every other unit (newton, joule, pascal, …) is derived from these seven.

Common dimensional formulae

Quantity	Dimensions
Velocity	$[LT^{-1}]$
Acceleration	$[LT^{-2}]$
Force	$[MLT^{-2}]$
Momentum	$[MLT^{-1}]$
Work / Energy	$[ML^2T^{-2}]$
Power	$[ML^2T^{-3}]$
Pressure	$[ML^{-1}T^{-2}]$
Density	$[ML^{-3}]$

Significant figures — the essentials

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant; trailing zeros after a decimal point are.
• In multiplication or division, keep as many significant figures as the least precise factor.

Errors

• Absolute error $= |\,\text{measured} - \text{true}\,|$.
• Relative error $= \dfrac{\Delta a}{a}$, and percentage error $= \dfrac{\Delta a}{a}\times 100\%$.
• For a product/quotient $x = \dfrac{a^p\, b^q}{c^r}$, the maximum relative error adds up as $\dfrac{\Delta x}{x} = p\dfrac{\Delta a}{a} + q\dfrac{\Delta b}{b} + r\dfrac{\Delta c}{c}$.

Beyond-NCERT JEE Formulae

A rapid-reference sheet of the high-yield results JEE Main leans on -- each with a one-line when to use and a [JEE Tip] for the trap that catches most students. (Basic SI units, standard dimensional formulae and elementary significant-figure rules are assumed from the earlier sections.)

1. Combination of errors -- the master formula

For a quantity built as $f=\dfrac{A^a B^b}{C^c}$, the maximum relative error is the weighted sum $$\dfrac{\Delta f}{f}=a\dfrac{\Delta A}{A}+b\dfrac{\Delta B}{B}+c\dfrac{\Delta C}{C}$$ When to use: any product or quotient of powers -- density, Young's modulus, resistivity, viscosity. Every exponent multiplies its own fractional error, and all terms add in magnitude.

[JEE Tip] Powers dominate the error budget. A quantity under a fourth power (like $r$ in Poiseuille's $\eta\propto r^4$) contributes four times its error, so an innocent-looking term is often the largest source. Never let signs cancel -- take every term positive to get the maximum error.

2. Sum and difference errors

For $Q=A\pm B$ the absolute errors add: $\Delta Q=\Delta A+\Delta B$. When to use: any sum or difference of measured quantities -- a net mass, or a length that is itself a difference of two scale readings.

[JEE Tip] The difference of two nearly equal numbers is a trap: $\Delta Q$ stays fixed while $Q$ shrinks, so the relative error explodes. This is why a small extension, or a real-minus-apparent depth, carries a huge percentage error.

3. Error when a quantity is raised to a power

If $Z=A^n$ then $\dfrac{\Delta Z}{Z}=|n|\dfrac{\Delta A}{A}$, valid for any real $n$ including $\tfrac12$ and $-1$. When to use: square roots (period $\propto\sqrt{L}$), reciprocals, and cubes (volume $\propto d^3$).

[JEE Tip] For $r=\dfrac{d}{2}$, $\dfrac{\Delta r}{r}=\dfrac{\Delta d}{d}$ -- halving the value does not halve the fractional error. For an area $\propto d^2$, $\dfrac{\Delta A}{A}=2\dfrac{\Delta d}{d}$.

4. Least count and instrument reading

Instrument	Least count
Vernier callipers	$1\,\text{MSD}-1\,\text{VSD}=\dfrac{1\,\text{MSD}}{N}$ when $N$ VSD $=(N-1)$ MSD
Screw gauge / micrometer	$\dfrac{\text{pitch}}{\text{no. of circular divisions}}$, with pitch $=\dfrac{\text{distance moved}}{\text{no. of rotations}}$

Reading $=\text{MSR}+(\text{LC}\times\text{coinciding division})\pm\text{zero-error correction}$. When to use: every callipers or screw-gauge numerical.

[JEE Tip] Get the sign right: a positive zero error is subtracted and a negative zero error is added, i.e. corrected $=$ observed $-$ (zero error, with its sign). Watch phrases like "the spindle advances 1 mm in 2 rotations" -- that makes the pitch 0.5 mm and halves the least count.

5. Dimensional formulae of important constants

Derive any of these on the spot: isolate the constant in its defining equation, then match dimensions.

Constant	From	Dimensions
Gravitational $G$	$F=\dfrac{Gm_1m_2}{r^2}$	$[M^{-1}L^3T^{-2}]$
Planck $h$	$E=h\nu$	$[ML^2T^{-1}]$
Boltzmann $k$	$E=\tfrac32 k\theta$	$[ML^2T^{-2}K^{-1}]$
Gas constant $R$	$PV=nRT$	$[ML^2T^{-2}K^{-1}\text{mol}^{-1}]$
Stefan $\sigma$	$P=\sigma A\theta^4$	$[MT^{-3}K^{-4}]$
Permittivity $\varepsilon_0$	$F=\dfrac{q_1q_2}{4\pi\varepsilon_0 r^2}$	$[M^{-1}L^{-3}T^4A^2]$
Permeability $\mu_0$	$\dfrac{F}{l}=\dfrac{\mu_0 I_1 I_2}{2\pi d}$	$[MLT^{-2}A^{-2}]$
Viscosity $\eta$	$F=6\pi\eta r v$	$[ML^{-1}T^{-1}]$
Surface tension $S$	$S=\dfrac{F}{l}$	$[MT^{-2}]$

When to use: "find the dimensions of X" and conversion-of-units problems.

[JEE Tip] Learn the twins: $h$ matches angular momentum $[ML^2T^{-1}]$; $RC$ and $\dfrac{L}{R}$ each give $[T]$; and $\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$ comes out as a speed -- it is $c$. Same-dimension pairs are favourite MCQ bait.

6. Dimensional analysis -- check and derive

• Homogeneity check: both sides of an equation, and every additive term within it, must carry the same dimensions -- use it to reject a wrong formula in seconds.
• Deduce a relation: write $Q=k\,x^a y^b z^c$, match the powers of $M$, $L$ and $T$ separately, and solve for $a$, $b$ and $c$.

[JEE Tip] The arguments of $\sin$, $\cos$, $e^x$ and $\ln$ are always dimensionless -- that one rule instantly fixes decay constants and wave numbers. Dimensions can never deliver a pure number (the $2$ in $\sqrt{2GM/R}$, the $\tfrac12$ in $\tfrac12 mv^2$), and the method fails when three or more terms could combine.

7. Significant figures and rounding

• Multiplication or division: keep as many significant figures as the factor with the fewest.
• Addition or subtraction: keep as many decimal places as the term with the fewest.
• Rounding a trailing 5: round so that the last kept digit is even ($2.745\to2.74$, $2.735\to2.74$).

When to use: reporting a final answer -- and round only at the very last step.

[JEE Tip] Exact numbers (a count of oscillations, the $2$ in $2\pi r$) carry infinite significant figures and never limit the result, and simply changing units never changes the significant-figure count.

Solved Examples -- Beyond-NCERT Formulae

Example 1: Maximum error in the density of a sphere.

A small sphere has mass $m=22.0\pm0.2$ g and diameter $d=2.00\pm0.01$ cm. Find its density and the maximum error in it.

Solution:

1. Density $\rho=\dfrac{m}{V}=\dfrac{m}{\tfrac43\pi r^3}=\dfrac{6m}{\pi d^3}$, since $r=\dfrac{d}{2}$.
2. Central value: $\rho=\dfrac{6(22.0)}{\pi(2.00)^3}=\dfrac{132}{8\pi}=5.25$ g/cm$^3$.
3. As a product of powers the fractional errors add, with the diameter's exponent tripled: $\dfrac{\Delta\rho}{\rho}=\dfrac{\Delta m}{m}+3\dfrac{\Delta d}{d}$.
4. $\dfrac{\Delta m}{m}=\dfrac{0.2}{22.0}=0.91\%$ and $3\dfrac{\Delta d}{d}=3\times\dfrac{0.01}{2.00}=1.5\%$, so $\dfrac{\Delta\rho}{\rho}=0.91+1.5=2.4\%$.
5. $\Delta\rho=0.024\times5.25=0.13$ g/cm$^3$.
6. Result: $\rho=(5.25\pm0.13)$ g/cm$^3$. The diameter dominates the error even though it was measured finely, because it enters cubed.

Example 2: Absolute error in a resistance.

Ohm's law gives $R=\dfrac{V}{I}$ with $V=100\pm5$ V and $I=10.0\pm0.2$ A. Report $R$ with its error.

Solution:

1. Central value: $R=\dfrac{100}{10.0}=10.0\ \Omega$.
2. For a quotient the fractional errors add: $\dfrac{\Delta R}{R}=\dfrac{\Delta V}{V}+\dfrac{\Delta I}{I}$.
3. $\dfrac{\Delta V}{V}=\dfrac{5}{100}=5\%$ and $\dfrac{\Delta I}{I}=\dfrac{0.2}{10.0}=2\%$, so $\dfrac{\Delta R}{R}=7\%$.
4. $\Delta R=0.07\times10.0=0.7\ \Omega$.
5. Result: $R=(10.0\pm0.7)\ \Omega$. The errors add even though $I$ sits in the denominator -- for a maximum-error estimate they never cancel.

Example 3: Screw-gauge diameter with a positive zero error.

A screw gauge has pitch 1 mm and 100 divisions on its circular scale. With the jaws closed the 4th circular division sits on the reference line -- a positive zero error. Gripping a wire, the main scale reads 2 mm and the 65th division coincides. Find the true diameter.

Solution:

1. Least count $=\dfrac{\text{pitch}}{\text{divisions}}=\dfrac{1}{100}=0.01$ mm.
2. Zero error $=+4\times0.01=+0.04$ mm (positive, so it must be subtracted).
3. Observed reading $=\text{MSR}+\text{CSR}\times\text{LC}=2+65\times0.01=2.65$ mm.
4. True diameter $=$ observed $-$ zero error $=2.65-0.04=2.61$ mm.
5. Skipping the zero correction would wrongly give 2.65 mm -- the single commonest slip in screw-gauge problems.

Example 4: Vernier reading with a negative zero error.

A vernier callipers has 10 vernier divisions equal to 9 main-scale divisions, with 1 MSD $=1$ mm. With the jaws closed the 7th vernier division coincides -- a negative zero error. Measuring a rod, the main scale reads 15 mm and the 4th vernier division coincides. Find the rod's length.

Solution:

1. $1\,\text{VSD}=\dfrac{9}{10}=0.9$ mm, so $\text{LC}=1\,\text{MSD}-1\,\text{VSD}=1-0.9=0.1$ mm.
2. Negative zero error $=-(N-p)\times\text{LC}=-(10-7)\times0.1=-0.3$ mm.
3. Observed reading $=15+4\times0.1=15.4$ mm.
4. Corrected $=$ observed $-$ zero error $=15.4-(-0.3)=15.7$ mm.
5. A negative zero error is effectively added back -- the usual sign slip is to subtract it instead.

Example 5: Using dimensions to reject a wrong formula.

A student cannot recall whether the escape speed is $v=\sqrt{\dfrac{2GM}{R}}$ or $v=\sqrt{\dfrac{2GM}{R^2}}$. Settle it by dimensions, given $[G]=[M^{-1}L^3T^{-2}]$.

Solution:

1. A speed has dimensions $[LT^{-1}]$.
2. First form: $\left[\dfrac{GM}{R}\right]=\dfrac{(M^{-1}L^3T^{-2})(M)}{L}=[L^2T^{-2}]$, and $\sqrt{[L^2T^{-2}]}=[LT^{-1}]$. This is a speed.
3. Second form: $\left[\dfrac{GM}{R^2}\right]=\dfrac{L^3T^{-2}}{L^2}=[LT^{-2}]$, and $\sqrt{[LT^{-2}]}=[L^{1/2}T^{-1}]$. This is not a speed.
4. Only $v=\sqrt{\dfrac{2GM}{R}}$ is dimensionally admissible. Dimensions verify the form but cannot supply the pure number 2.

Example 6: Dimensional formula of a physical constant.

Find the dimensional formula and SI unit of the gravitational constant $G$ from Newton's law $F=\dfrac{Gm_1m_2}{r^2}$.

Solution:

1. Isolate the constant: $G=\dfrac{Fr^2}{m_1m_2}$.
2. Substitute dimensions: $[G]=\dfrac{[MLT^{-2}][L^2]}{[M][M]}$.
3. Numerator $=[ML^3T^{-2}]$; denominator $=[M^2]$.
4. $[G]=\dfrac{ML^3T^{-2}}{M^2}=[M^{-1}L^3T^{-2}]$.
5. SI unit: N m$^2$ kg$^{-2}$, equivalently m$^3$ kg$^{-1}$ s$^{-2}$.

Example 7: Percentage error in $g$ from a pendulum.

In $g=\dfrac{4\pi^2 L}{T^2}$ the length is $L=100.0$ cm on a scale of least count 0.1 cm, and the time for 100 oscillations is 200.0 s on a stopwatch of least count 0.1 s. Find $g$ with its error.

Solution:

1. $T=\dfrac{200.0}{100}=2.000$ s and $L=1.000$ m, so $g=\dfrac{4\pi^2(1.000)}{(2.000)^2}=\pi^2=9.87$ m/s$^2$.
2. $\%L=\dfrac{0.1}{100.0}=0.1\%$; and $\dfrac{\Delta T}{T}=\dfrac{\Delta t}{t}=\dfrac{0.1}{200.0}=0.05\%$, since dividing the total time by 100 leaves the fractional error unchanged.
3. Because $g\propto L\,T^{-2}$: $\%g=\%L+2\,\%T=0.1+2(0.05)=0.2\%$.
4. $\Delta g=0.002\times9.87=0.02$ m/s$^2$.
5. Result: $g=(9.87\pm0.02)$ m/s$^2$. The crude timing, acting through the $T^2$, governs the error -- not the length.

Example 8: Significant figures and the round-to-even rule.

(a) A calculator returns $2.5\times3.42=8.55$. (b) Two intermediate results are 2.745 and 2.735. Report (a) to the justified significant figures, and round both numbers in (b) to 3 significant figures.

Solution:

1. (a) In multiplication the answer keeps the fewest significant figures among the factors: 2.5 has 2 and 3.42 has 3, so keep 2.
2. Rounding $8.55$ to 2 significant figures: the digit dropped is 5, and the convention is to make the last kept digit even, giving $8.55\to8.6$.
3. (b) $2.745\to2.74$: on dropping the 5, the preceding digit 4 is already even, so it stays.
4. $2.735\to2.74$: on dropping the 5, the preceding digit 3 is odd, so it rounds up to the even 4.
5. Both settle on 2.74; rounding half to even removes the slight upward bias of always rounding a 5 up.