Units and Measurements Formula Sheet — JEE Main Physics

Every key Units and Measurements formula, definition and theorem for JEE Main Physics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 10 sub-topics

Units of measurement
System of units
SI units
Fundamental and derived units
Least count
Accuracy and precision of measuring instruments
Errors in measurement
Significant figures
Dimensions of physical quantities
Dimensional analysis and its applications

Units, Systems & the SI

Measurement:

Q = n u

— a physical quantity

=

a pure number n

\times

a unit u. For a fixed Q,

n u =

const

\Rightarrow n \propto \frac{1}{u}

smaller

unit gives a

larger

number).

Base vs derived (self-contained)

▸ $Base$ (fundamental) quantities are mutually independent: length, mass, time, current, temperature, amount of substance, luminous intensity ( $7$ ).
▸ $Derived$ quantities $=$ products/powers of base ones; their units are $derived units$ (e.g. $m s^{- 1}$ ).
▸Base $+$ derived units form a $system$ : CGS, FPS (British), MKS, $SI$ . SI (1971, CGPM) is coherent (no conversion factors) and decimal.

Seven SI base units

Quantity	Unit	Sym	Quantity	Unit	Sym
Length	metre	m	Temp.	kelvin	K
Mass	kilogram	kg	Amount	mole	mol
Time	second	s	Lum. int.	candela	cd
Current	ampere	A	—	—	—

Base-unit definitions (SI since 2019: exact constants)

▸ $second$ : Cs- $133$ hyperfine frequency $Δ ν_{Cs} = 9, 192, 631, 770$ Hz. $metre$ : $c = 299, 792, 458 m s^{- 1}$ (path of light in $1/299, 792, 458$ s).
▸ $kilogram$ : $h = 6.62607015 \times 1 0^{- 34}$ J s. $ampere$ : $e = 1.602176634 \times 1 0^{- 19}$ C.
▸ $kelvin$ : $k_{B} = 1.380649 \times 1 0^{- 23} J K^{- 1}$ . $mole$ : $N_{A} = 6.02214076 \times 1 0^{23} mol^{- 1}$ . $candela$ : $683 lm W^{- 1}$ at $540 \times 1 0^{12}$ Hz.
▸Old (pre-2019) wording still met in books: kg $=$ Paris prototype; K $= \frac{1}{273.16}$ of water's triple point; mol $=$ entities in $0.012$ kg of $^{12} C$ .

Supplementary (dimensionless) units

▸Plane angle $d θ = \frac{d s}{r}$ in $radian$ ; solid angle $d Ω = \frac{d A}{r ^{2}}$ in $steradian$ . $d s =$ arc, $d A =$ area on a sphere of radius r. Both have no dimension.

🚫 Examiner Trap · Units & systems

(1)

n \propto 1/ u

— bigger unit

\Rightarrow

smaller

number; students wrongly scale the same way. (2) candela

=

luminous

intensity

, not brightness/flux; mole

=

amount, not mass. (3) Angle/steradian are

units but not

base quantities (they're dimensionless).

SI prefixes

Prefix	Sym	Factor	Prefix	Sym	Factor
exa	E	$1 0^{18}$	deci	d	$1 0^{- 1}$
peta	P	$1 0^{15}$	centi	c	$1 0^{- 2}$
tera	T	$1 0^{12}$	milli	m	$1 0^{- 3}$
giga	G	$1 0^{9}$	micro	$μ$	$1 0^{- 6}$
mega	M	$1 0^{6}$	nano	n	$1 0^{- 9}$
kilo	k	$1 0^{3}$	pico	p	$1 0^{- 12}$
hecto	h	$1 0^{2}$	femto	f	$1 0^{- 15}$
deca	da	$1 0^{1}$	atto	a	$1 0^{- 18}$

Named SI derived units

Quantity	Unit	In base units
Frequency	hertz (Hz)	$s^{- 1}$
Force	newton (N)	$kg m s^{- 2}$
Energy/Work	joule (J)	$kg m^{2} s^{- 2}$
Power	watt (W)	$kg m^{2} s^{- 3}$
Pressure	pascal (Pa)	$kg m^{- 1} s^{- 2}$
Charge	coulomb (C)	$A s$
Potential	volt (V)	$kg m^{2} s^{- 3} A^{- 1}$
Resistance	ohm ( $Ω$ )	$kg m^{2} s^{- 3} A^{- 2}$
Mag. flux	weber (Wb)	$kg m^{2} s^{- 2} A^{- 1}$
Mag. field	tesla (T)	$kg s^{- 2} A^{- 1}$

🚫 Examiner Trap · Writing units (NCERT rules)

No plural, no full stop:

10 kg

(not

10

kgs.). Unit named after a person

\to

capital symbol

(N, Pa, Hz, W) but

lowercase name

(newton, pascal). Always leave a space between number and unit:

5 N

Practical Units & Scales of Nature

Practical units (length, mass, energy)

Unit	Value	Unit	Value
fermi (f)	$1 0^{- 15} m$	$angstrom \overset{˚}{A}$	$1 0^{- 10} m$
AU	$1.496 \times 1 0^{11} m$	light year (ly)	$9.46 \times 1 0^{15} m$
parsec	$3.08 \times 1 0^{16} m$		$= 3.26 ly$
a.m.u. (u)	$1.66 \times 1 0^{- 27} kg$		$= 931.5$ $MeV / c^{2}$
eV	$1.6 \times 1 0^{- 19} J$	quintal / tonne	$1 0^{2}$ / $1 0^{3} kg$

What each unit measures

▸ $Length$ : fermi, Å, AU, ly, parsec. AU $=$ mean Sun–Earth distance; ly $=$ distance light travels in $1$ year; parsec $=$ distance at which $1$ AU subtends $1^{''}$ ( $1$ arc-second).
▸ $Mass$ : u (a.m.u.), quintal, tonne; $1 u = \frac{1}{12}$ mass of a $_{6}^{12} C$ atom.
▸ $Energy$ : eV $=$ energy gained by an electron across $1$ V.

🚫 Examiner Trap · Practical units

Light year

and

parsec

are

distances

not time

(classic trap). Size order:

1 parsec > 1 ly > 1 AU

. Do not mix

1 AU \neq = 1 ly

1^{''}

(arc-sec),

1^{'}

(arc-min):

1^{\circ} = 6 0^{'} = 360 0^{''}

Units retained for use (NCERT Table 2.2)

Unit	Value	Unit	Value
bar	$1 0^{5} Pa$	atm	$1.013 \times 1 0^{5} Pa$
litre	$1 0^{- 3} m^{3}$	degree	$\frac{π}{180} rad$
carat	$200 mg$	barn	$1 0^{- 28} m^{2}$
hectare	$1 0^{4} m^{2}$	year	$3.156 \times 1 0^{7} s$

Order of magnitude: Write

Q = a \times 1 0^{b}

with

1 \leq a < 10

. Order

= b

a \leq 5

, else

b + 1

(NCERT threshold

5

). It is the power of

10

nearest

to Q.

🚫 Examiner Trap · Order of magnitude

Forget to first normalise to

1 \leq a < 10

\Rightarrow

wrong b (e.g.

0.0006 = 6 \times 1 0^{- 4}

, order

= - 3

, not

- 4

). NCERT uses the

a \leq 5

cut; some books use

a \leq 10 \approx 3.16

— for JEE, follow the

5

rule unless told otherwise.

Orders of magnitude (NCERT Tables 2.3–2.5)

Length (m)	Mass (kg)	Time (s)
proton $1 0^{- 15}$	electron $1 0^{- 30}$	light wave $1 0^{- 15}$
atom $1 0^{- 10}$	proton $1 0^{- 27}$	atomic vib. $1 0^{- 15}$
virus $1 0^{- 8}$	blood cell $1 0^{- 13}$	heartbeat $1 0^{0}$
human $1 0^{0}$	human $1 0^{2}$	day $1 0^{5}$
Earth R $1 0^{7}$	Earth $1 0^{25}$	human life $1 0^{9}$
Sun–Earth $1 0^{11}$	Sun $1 0^{30}$	Egypt pyr. $1 0^{11}$
galaxy $1 0^{21}$	galaxy $1 0^{41}$	dinosaurs $1 0^{15}$
universe $1 0^{26}$	universe $1 0^{55}$	universe $1 0^{17}$

Estimation — size of a molecule (oleic acid)

▸Mono-layer thickness $t = \frac{volume of acid}{area of film} = \frac{nV}{20 \times 20 A} \approx 1 0^{- 9}$ m (NCERT 2.3.2). Acid is diluted $1 : 20$ twice, so n drops of solution (volume V each) carry $\frac{nV}{400}$ of acid; $A =$ area of the film they spread into.

⚡ Shortcut · Curious large number

Ratio of largest

/

smallest lengths

\approx 1 0^{41}

— same order as longest

/

shortest times. A recurring 'large number' in nature (NCERT).

Dimensional Formulae

Dimensions: The powers to which the base quantities [M],[L],[T],[A],[K] are raised in a quantity, e.g.

[F] = [M^{1} L^{1} T^{- 2}]

. They are

independent

of the formula chosen (

F = ma

F = GmM / r^{2}

give the same).

Symbols (define before plugging)

▸ $M =$ mass, $L =$ length, $T =$ time, $A =$ current, $K =$ temperature, $mol =$ amount, $cd =$ luminous intensity.

Mechanics, fluids & gravitation

Quantity	Dim. formula
Area / Volume	$[L^{2}]$ / $[L^{3}]$
Velocity / Acceleration	$[L T^{- 1}]$ / $[L T^{- 2}]$
Force / Weight	$[MLT^{- 2}]$
Momentum / Impulse	$[MLT^{- 1}]$
Work / Energy / Torque	$[M L^{2} T^{- 2}]$
Power	$[M L^{2} T^{- 3}]$
Pressure / Stress / Modulus	$[M L^{- 1} T^{- 2}]$
Density	$[M L^{- 3}]$
Frequency / $ω$ / decay const	$[T^{- 1}]$
Angular momentum	$[M L^{2} T^{- 1}]$
Moment of inertia	$[M L^{2}]$
Grav. constant $G$	$[M^{- 1} L^{3} T^{- 2}]$
Grav. potential	$[L^{2} T^{- 2}]$
Surface tension / Spring $k$	$[M T^{- 2}]$
Coeff. of viscosity $η$	$[M L^{- 1} T^{- 1}]$
Planck const $h$	$[M L^{2} T^{- 1}]$

Comparative: quantities that SHARE dimensions

Dimension	Quantities with this dimension
$[M L^{2} T^{- 2}]$	Work, Energy, Torque, Heat, $P V$ , $RT$
$[M L^{- 1} T^{- 2}]$	Pressure, Stress, Modulus, Energy density, $\frac{1}{2} ε_{0} E^{2}$
$[MLT^{- 1}]$	Momentum, Impulse
$[M L^{2} T^{- 1}]$	Angular momentum, Planck const $h$
$[M T^{- 2}]$	Surface tension, Spring const, Force gradient
$[T^{- 1}]$	Frequency, $ω$ , velocity gradient, decay const, $R / L$
$[T]$	$RC, L / R, L C$ , time period

Electromagnetism, heat & modern

Quantity	Dim. formula
Charge $q$	$[A T]$
Potential / EMF $V$	$[M L^{2} T^{- 3} A^{- 1}]$
Electric field $E$	$[MLT^{- 3} A^{- 1}]$
Resistance $R$	$[M L^{2} T^{- 3} A^{- 2}]$
Resistivity $ρ$	$[M L^{3} T^{- 3} A^{- 2}]$
Capacitance $C$	$[M^{- 1} L^{- 2} T^{4} A^{2}]$
Inductance $L$	$[M L^{2} T^{- 2} A^{- 2}]$
Magnetic field $B$	$[M T^{- 2} A^{- 1}]$
Magnetic flux $ϕ$	$[M L^{2} T^{- 2} A^{- 1}]$
Permittivity $ε_{0}$	$[M^{- 1} L^{- 3} T^{4} A^{2}]$
Permeability $μ_{0}$	$[MLT^{- 2} A^{- 2}]$
Specific heat $c$	$[L^{2} T^{- 2} K^{- 1}]$
$k_{B}$ / Entropy	$[M L^{2} T^{- 2} K^{- 1}]$
Gas constant $R$	$[M L^{2} T^{- 2} K^{- 1} mol^{- 1}]$
Stefan const $σ$	$[M T^{- 3} K^{- 4}]$
Wien const $b$	$[L K]$

⚡ Shortcut · Useful constant combinations

RC = \frac{L}{R} = L C = [T]

μ_{0} ε_{0} = \frac{1}{c ^{2}} = [L^{- 2} T^{2}]

\frac{1}{2} ε_{0} E^{2} = \frac{B ^{2}}{2 μ _{0}} = [M L^{- 1} T^{- 2}]

(energy density).

h c = [M L^{3} T^{- 2}]

Planck units (JEE favourite):

\frac{h G}{c ^{3}} = [L]

\frac{h G}{c ^{5}} = [T]

\frac{h c}{G} = [M]

\frac{h c ^{5}}{G} = [M L^{2} T^{- 2}]

(energy).

\frac{e ^{2}}{4 π ε _{0} ℏ c}

is dimensionless (

\approx \frac{1}{137}

, fine-structure constant).

Dimensionless quantities (no

M, L, T

)

▸Angle, solid angle, strain, refractive index, relative density, friction coeff. $μ$ , Poisson's ratio, dielectric const., all pure ratios.
▸Arguments of $sin, cos, tan, lo g, e^{x}$ — and the functions' values — are dimensionless.

🚫 Examiner Trap · Dimensional formulae

(1) Same dimension

\neq =

same quantity: Work, Torque & Energy all

[M L^{2} T^{- 2}]

but are physically different. (2) Strain,

μ

, refractive index, angle are

dimensionless

— don't assign [L] etc. (3)

[G] \neq = [g]

[G] = [M^{- 1} L^{3} T^{- 2}]

[g] = [L T^{- 2}]

. (4) Use [A] (current),

not

charge, as the EM base dimension.

Dimensional Analysis & Applications

Principle of homogeneity:Every additive term of a physically correct equation has the

same

dimensions; only like quantities add or subtract.

Necessary, not sufficient

for correctness.

Three uses

▸ $Check$ : in $x = x_{0} + v_{0} t + \frac{1}{2} a t^{2}$ each term $= [L]$ ✓.
▸ $Derive$ a formula: assume a product of powers, match M,L,T.
▸ $Convert$ a value between unit systems.

⚡ Shortcut · Find an unknown's dimension

A term adds only to a like term. In

(L + α)

[α] = [L]

. In

(F - β)

[β] = [MLT^{- 2}]

. Inside

sin (βt), lo g, e^{x}

the

argument

dimensionless

\Rightarrow [β] = [T^{- 1}]

, and the whole function is dimensionless.

✎ Example · Find

[α], [β]

α = \frac{F}{v ^{2}} sin (βt)

—

F =

force,

v =

velocity,

t =

time

$βt$ dimensionless $\Rightarrow [β] = [T^{- 1}]$ ; $sin (\cdot)$ dimensionless $\Rightarrow [α] = \frac{[ M L T ^{- 2} ]}{[ L T ^{- 1} ] ^{2}} = [M L^{- 1} T^{0}]$

∴

[α] = [M L^{- 1}], [β] = [T^{- 1}]

Time period of a pendulum (NCERT Ex 2.17)

$T \propto l^{x} g^{y} m^{z} \Rightarrow$ $M : z = 0,$ $L : x + y = 0,$ $T : - 2 y = 1$ — $x = \frac{1}{2}, y = - \frac{1}{2}, z = 0$ (l length, g accn, m mass)
$T = k l / g$ — experiment gives $k = 2 π$ (dimensionless, $not$ derivable)

🚫 Examiner Trap · Limitations of dimensional analysis

Cannot: (1) find dimensionless constants (

2 π, \frac{1}{2}, k

); (2) derive relations with

+

-

> 1

term; (3) handle

sin / cos / lo g / e^{x}

relations; (4) solve when

> 3

unknowns (only

3

eqns from M,L,T).

Dimensionally correct

\neq =

physically correct

, but dimensionally WRONG

=

definitely wrong.

Unit conversion

Convert a value

n_{2} = n_{1} [\frac{M _{1}}{M _{2}}]^{a} [\frac{L _{1}}{L _{2}}]^{b} [\frac{T _{1}}{T _{2}}]^{c}

n_{1}, n_{2} =

numbers in old/new units;

a, b, c =

dimensions of the quantity in M,L,T; subscript

1 =

old system,

2 =

new

Worked conversions

▸ $1 N = \frac{kg m}{s ^{2}}$ $= (1 0^{3} g) (1 0^{2} cm) s^{- 2}$ $= 1 0^{5} dyne$ .
▸ $G = 6.67 \times 1 0^{- 11} SI = 6.67 \times 1 0^{- 8} CGS$ (c $m^{3}$ $g^{- 1}$ $s^{- 2}$ ).
▸ $ρ = 2 g/cm^{3} = 2 \times 1 0^{3} kg/m^{3}$ . km/h $\to$ m/s: $\times \frac{5}{18}$ ; m/s $\to$ km/h: $\times \frac{18}{5}$ .

⚡ Shortcut · New fundamental quantities

If V,F,T are taken as fundamental:

[M] = [V^{- 1} F T]

[E] = [V F T]

. Method: write

Q = V^{a} F^{b} T^{c}

, substitute known dimensions of V,F,T in M,L,T, and match.

Dimensionless numbers (fluid mechanics)

▸Reynolds number $R e = \frac{ρ v D}{η}$ (inertial $/$ viscous force ratio); $ρ$ density, v speed, D pipe dia, $η$ viscosity.
▸Stokes drag $F = 6 π ηr v$ ; the $6 π$ is a pure number dimensional analysis cannot give.

🚫 Examiner Trap · Conversion & homogeneity

(1) Use

old / new

ratio consistently — inverting

M_{1} / M_{2}

is the #1 slip. (2) A wrong dimensionless constant still passes the homogeneity check — so checking can't

prove

a formula. (3) Angles must be in

radian

before using

sin θ \approx θ

Significant Figures, Rounding & Errors

Significant figures: All reliable digits

+

the first uncertain digit; they reflect precision (the least count). A change of

unit

never changes the count (

4.700 m = 470.0 cm

, both

4

s.f.).

Counting rules

▸All non-zero digits, and zeros $between$ them, are significant.
▸ $Leading$ zeros are NOT significant ( $0.00418 \to 3$ ). $Trailing$ zeros after a decimal ARE ( $3.0800 \to 5$ ).
▸Plain-integer trailing zeros are ambiguous ( $3500 \to$ write as $a \times 1 0^{b}$ ). Exact numbers (the $2$ in $d /2$ , count n in $T = t / n$ ) have $infinite$ s.f.

Comparative: s.f. arithmetic

Operation	Keep	Example
$\times$ / $\div$	fewest $s.f.$	$\frac{4.237}{2.51} = 1.69$ (3 s.f.)
$+$ / $-$	fewest $decimal places$	$436.32 + 227.2$ $+ 0.301$ $= 663.8$

Rounding off (round-to-even)

▸Drop $< 5$ → keep last digit ( $2.743 \to 2.74$ ); drop $> 5$ → raise ( $2.746 \to 2.75$ ).
▸Drop exactly $5$ (nothing after) → make the last kept digit $even$ ( $2.745 \to 2.74$ , $2.735 \to 2.74$ ).

🚫 Examiner Trap · Significant figures

(1) Changing units does NOT change s.f. count —

0.0250

kg and

25.0

g both have 3. (2) Round only the

final

answer, never intermediates. (3) Trailing zeros without a decimal are ambiguous —

1500

may be

2, 3

4

s.f.

Accuracy vs Precision (comparative) — not in JEE Main 2026 syllabus

	Accuracy	Precision
Means	Close to TRUE value	Reproducible / tight cluster
Limited by	Systematic error	Random error & least count
Improve via	Calibration, zero correction	Averaging many readings

Four dartboards: high/low accuracy vs precision — Accuracy (to centre) vs precision (grouping) — they are independent.

Types of error (comparative)

Type	Nature	Cure
Systematic	one fixed sign (zero error, technique, personal)	correct / calibrate
Random	irregular both signs	average many trials
Least-count	$=$ resolution of instrument	finer instrument
Gross	blunder / mistake	repeat carefully

Error propagation

Mean & abs. error

a_{mean} = \frac{\sum a _{i}}{n},

Δ a_{mean} = \frac{\sum ∣ a _{mean} - a _{i} ∣}{n}

a_{i} =

reading i,

n =

no. of readings;

Δ a =

mean absolute error

Relative / %

δ a = \frac{Δ a _{mean}}{a _{mean}},

% err = δ a \times 100

δ a =

relative (fractional) error

Result $Z$	Error rule
$Z = A \pm B$	$Δ Z = Δ A + Δ B$ (abs. errors ADD)
$Z = \frac{A ^{p} B ^{q}}{C ^{r}}$	$\frac{Δ Z}{Z} = ∣ p ∣ \frac{Δ A}{A}$ $+ ∣ q ∣ \frac{Δ B}{B}$ $+ ∣ r ∣ \frac{Δ C}{C}$

Worked (errors always ADD)

▸ $R = \frac{V}{I}$ : $5% + 2% = 7%$ . $g = \frac{4 π ^{2} L}{T ^{2}}$ : $\frac{Δ g}{g} = \frac{Δ L}{L} + 2 \frac{Δ T}{T}$ .
▸ $ρ = \frac{m}{π r ^{2} ℓ}$ : $\frac{Δ ρ}{ρ} = \frac{Δ m}{m} + 2 \frac{Δ r}{r} + \frac{Δ ℓ}{ℓ}$ (power of r is $2$ ).
▸Dependent $R = \frac{X Y}{X + Y}$ : differentiate first $\Rightarrow \frac{Δ R}{R ^{2}} = \frac{Δ X}{X ^{2}} + \frac{Δ Y}{Y ^{2}}$ .

🚫 Examiner Trap · Error propagation

(1) For

Z = A - B

the absolute errors still

ADD

(never subtract) and

%

error blows up for small

A - B

. (2) Multiply each fractional error by

∣ power ∣

— the

2

for

r^{2}

\frac{1}{2}

for

. (3) Max-permissible error

=

least count (place value of last digit). For n oscillations:

\frac{Δ T}{T} = \frac{LC}{total time}

— time MORE oscillations to cut it.

Measuring Instruments

Comparative: Vernier vs Screw gauge

	Vernier callipers	Screw gauge
Least count	$0.1$ mm	$0.01$ mm
Principle	$1 MS D - 1$ VSD	pitch / circular div.
LC formula	$\frac{1 MSD}{N}$	$\frac{pitch}{CSD}$
Reading	MSR $+ V S D \times$ LC	MSR $+ CS D \times$ LC

Vernier LC

LC = 1 MSD - 1 VSD = \frac{1 MSD}{N}

MSD

=

main-scale div.; VSD

=

vernier div.;

N

VSD coincide with

(N - 1)

MSD

Vernier reading

= MSR

+ (coinciding VSD) \times LC

- (zero error)

MSR

=

main-scale reading just left of the jaw

Vernier: 10 divisions span 9 mm; vernier 0 at 1.3 mm, vernier line 3 coincides with main-scale 4 — $10$ VSD $= 9$ mm $\Rightarrow$ LC $= 0.1$ mm. Reading $= 1 + 3 \times 0.1 = 1.3$ mm.

Vernier zero error: vernier 0 is 0.4 mm right of main 0 (vernier 4 coincides, +0.04 cm) or 0.4 mm left (vernier 6 coincides, -0.04 cm) — Positive ( $+ 0.04$ ) and negative ( $- 0.04$ cm) zero error.

Screw gauge LC

LC = \frac{pitch}{no. of circular divisions}

pitch

= \frac{dist. moved}{no. of rotations}

=

distance per full turn; CSD

=

circular-scale division

Screw gauge side view: sleeve with mm and half-mm marks, thimble edge at 3.77 mm, thimble division 27 on the datum line — Reading $=$ MSR $+$ CSR $\times$ LC $= 3.5 + 27 \times 0.01 = 3.77$ mm.

Zero-error correction

true = observed

- (zero error, with sign)

+

ve zero error

\Rightarrow

SUBTRACT;

-

\Rightarrow

ADD (subtracting a negative)

Screw gauge with jaws closed: thimble 0 below the datum line (+3 divisions) and above it (division 47, -3 divisions) — Jaws closed: $0$ below the line $\Rightarrow +$ error; $0$ above $\Rightarrow -$ error.

🚫 Examiner Trap · Instruments & zero error

(1) Correction:

always subtract

the

signed

zero error — a

negative

error is therefore ADDED. (2)

-

ve vernier error

= - (N - coinciding) \times

LC. (3) pitch

=

dist

/

rotations — don't read it as one division. (4) Account for

backlash

: turn the screw one way only while taking readings.

Large distances — parallax

Parallax

D = \frac{b}{θ}, α = \frac{d}{D}

D =

distance to object,

b =

basis (separation of observation points),

θ =

parallax angle (rad),

α =

angular size,

d =

object diameter

Parallax: object from A and B, basis b, angle theta — Parallax method.

🚫 Examiner Trap · Parallax & angles

θ

MUST be in radian. Conversions:

1^{\circ} = 1.745 \times 1 0^{- 2}

1^{'} = 2.909 \times 1 0^{- 4}

1^{''} = 4.848 \times 1 0^{- 6}

rad. Mixing degrees into

D = b / θ

is a frequent error.

Key Experiments & Error Analysis

g

by a simple pendulum

g = 4 π^{2} \frac{L}{T ^{2}}, \frac{Δ g}{g} = \frac{Δ L}{L} + 2 \frac{Δ T}{T}

L =

thread length

+ \frac{1}{2}

(bob dia.)

+

hook; plot L vs

T^{2}

, slope

= g /4 π^{2}

Pendulum and L vs T-squared straight line — L– $T^{2}$ line; slope gives g.

🚫 Examiner Trap · Pendulum

T dominates the error (

\times 2

) — time

many

oscillations, not one. Add HALF the bob diameter to the length.

T^{2} \propto L

, so the graph is a straight line through the origin, not T vs L.

Young's modulus — Searle's method

Y = \frac{M gL}{π r ^{2} ℓ}

M =

load mass,

L =

wire length,

r =

wire radius,

ℓ =

extension; plot

ℓ

vs load, slope

= L / (π r^{2} Y)

. Error:

\frac{Δ Y}{Y} = \frac{Δ M}{M} + \frac{Δ L}{L} + 2 \frac{Δ r}{r} + \frac{Δ ℓ}{ℓ}

Searle double-wire apparatus and extension vs load line — Reference + experimental wire; slope $= ℓ / M g$ .

🚫 Examiner Trap · Searle's

Y

r enters as

r^{2}

(measure with screw gauge — its small error doubles). The

two-wire

design cancels temperature expansion & support yielding; a single wire would give a systematic error.

Focal length —

u

–

v

method

Mirror:

\frac{1}{v} + \frac{1}{u} = \frac{1}{f}

; lens:

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}

; error:

\frac{Δ f}{f ^{2}} = \frac{Δ u}{u ^{2}} + \frac{Δ v}{v ^{2}}

Cartesian signs;

u =

object dist.,

v =

image dist.,

f =

focal length. For a real image both reduce to

\frac{1}{∣ v ∣} + \frac{1}{∣ u ∣} = \frac{1}{∣ f ∣}

: plot

1/∣ v ∣

1/∣ u ∣

— both intercepts

= 1/∣ f ∣

1/v vs 1/u straight line with intercepts 1/f — Intercepts give $1/∣ f ∣$ .

Speed of sound — resonance column

λ = 2 (l_{2} - l_{1}), v = f λ

l_{1}, l_{2} =

1st & 2nd resonance lengths,

f =

fork frequency; end-correction

e = 0.3 d

(

d =

tube dia.)

Resonance tube with tuning fork: l1 to the first-resonance water level (dashed), l2 to the second — Two resonances $\Rightarrow λ = 2 (l_{2} - l_{1})$ .

🚫 Examiner Trap · Optics & sound experiments

(1) u,v carry

sign

; plotting

1/ v

1/ u

removes the constant. (2) Taking

λ = 2 (l_{2} - l_{1})

cancels the end-correction

— using

λ = 4 l_{1}

alone does not. (3) Index/bench error: observed distance

\neq =

true; add index correction

=

(actual

-

observed) to every reading.

More JEE Main Physics formula sheets

Frequently Asked Questions

What are the most important Units and Measurements formulas for JEE Main?

This Units and Measurements formula sheet covers all the high-yield Physics formulas, definitions and theorems you need for JEE Main, across Units of measurement, System of units, SI units, Fundamental and derived units, Least count — each shown with the key result and, where useful, a worked example.

Is this Units and Measurements formula sheet free?

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How should I revise Units and Measurements formulas?

Blurt the Units and Measurements formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

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