Dimensional analysis — checking equations, deriving relations

WHAT are dimensions?

Examples built from first principles (always go back to the definition of the quantity):

Quantity	Defined as	Dimension
Velocity	displacement / time	$[L T^{-1}]$
Acceleration	velocity / time	$[L T^{-2}]$
Force	mass × acceleration	$[M L T^{-2}]$
Work / Energy	force × distance	$[M L^2 T^{-2}]$
Pressure	force / area	$[M L^{-1} T^{-2}]$
Power	energy / time	$[M L^2 T^{-3}]$

The Principle of Homogeneity (the engine)

WHY does this rule exist? Because addition only makes sense between identical kinds. "5 metres + 3 seconds" has no meaning — there is no number that is both. So if a formula adds two terms, nature forces them to be the same dimension.

Corollaries (memorise the consequences, not the lines):

Arguments of $\sin,\cos,\ln,e^x$ must be dimensionless (you can't take the sine of "2 metres").
Exponents and angles are dimensionless.
A pure number has dimension $[M^0L^0T^0]$ .

USE 1 — Checking an equation

USE 2 — Deriving a relation (the powerful trick)

Worked example Period of a simple pendulum

Guess: the period $T$ might depend on length $\ell$ , mass $m$ , gravity $g$ : $T = k\, \ell^{a} m^{b} g^{c}$ where $k$ is a dimensionless constant.

Step 1 — write dimensions. $[T]=T$ , $[\ell]=L$ , $[m]=M$ , $[g]=LT^{-2}$ . $T^1 = L^{a}\, M^{b}\, (LT^{-2})^{c} = M^{b}\,L^{a+c}\,T^{-2c}$

Why this step? We collected powers of each base separately, because each base is an independent "axis."

Step 2 — match powers of each base.

$M:\ b = 0$
$L:\ a + c = 0$
$T:\ -2c = 1 \Rightarrow c = -\tfrac12 \Rightarrow a = \tfrac12$

Step 3 — assemble. $T = k\,\ell^{1/2} g^{-1/2} = k\sqrt{\frac{\ell}{g}}$

Result & insight: $T \propto \sqrt{\ell/g}$ , and $T$ doesn't depend on mass — the method told us $b=0$ automatically. (The true constant is $k=2\pi$ , which dimensional analysis cannot supply.)

Limitations (know the boundaries)

80/20 — the few things that earn the marks

Flashcards

What does the dimension of a quantity tell you?

How it is built from base quantities

M,L,T

— its kind, independent of the chosen unit.

State the principle of homogeneity.

Every term added or equated in a valid equation must have identical dimensions.

Dimension of force?

[M L T^{-2}]

(from

F=ma

Dimension of energy/work?

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

(force × distance).

Dimension of pressure?

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

(force / area).

Dimension of power?

[M L^2 T^{-3}]

(energy / time).

Why must the argument of

\sin

e^x

be dimensionless?

These functions equal sums of powers of their argument; adding

x

x^2

requires

x

to be a pure number.

Can dimensional analysis prove an equation correct?

No — it's blind to pure numbers (like

\tfrac12

), signs in sums, and trig/exp forms; it can only disprove or fail to disprove.

A failed dimensional check means?

The equation is definitely wrong.

Derive how pendulum period scales with

\ell

and

g

T=k\,\ell^a g^c

; matching

L,T

gives

a=\tfrac12,\ c=-\tfrac12

, so

T\propto\sqrt{\ell/g}

Why is the pendulum period independent of mass?

Matching the power of

M

gives

b=0

since

g

and

\ell

carry no mass.

Three failures of the derivation method?

Can't find numeric constants; fails when unknowns > base dimensions; can't build multi-term (sum) formulas.

Convert 1 N to dyne using dimensions.

[F]=MLT^{-2}\Rightarrow 1N=10^3g\cdot10^2cm\cdot s^{-2}=10^5

dyne.

Recall Feynman: explain to a 12-year-old

Imagine every measurement wears a colored shirt: red for length, blue for time, green for weight. You're only allowed to add things wearing the same shirt — two red lengths give a red length, fine; but a red length plus a blue time is nonsense. When grown-ups write a physics equation, you can check it by looking at the shirts on both sides: if they match, the equation isn't obviously broken; if they clash, it's definitely wrong. The cool bonus: if you know what shirts the ingredients wear, you can often guess the recipe — like guessing a swing's wobble time grows with its length and shrinks with stronger gravity — just by making the shirts match up.

Connections

Units and the SI system — dimensions vs the units that realize them.
Significant figures and error propagation — numerical side of measurement.
Equations of motion (kinematics) — every result is dimensionally checkable.
Buckingham Pi theorem — the general formalism behind the derivation trick.
Vectors — components and addition — components share the dimension of the vector.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, har physical quantity ka ek "type" hota hai — koi length hai, koi time, koi mass. Inko hum dimensions kehte hain: $[L]$ , $[T]$ , $[M]$ . Asli rule bahut simple hai — sirf same type ki cheezein add ho sakti hain. 5 metre aur 3 second ko jod nahi sakte, kyunki dono alag-alag kind ke hain. Isi rule ko principle of homogeneity bolte hain: equation ke dono side, aur jo bhi terms plus/minus se jude hain, sabki dimension same honi chahiye.

Pehla use: equation check karna. Jaise $v^2 = u^2 + 2as$ — teeno terms ka dimension $L^2T^{-2}$ nikalta hai, toh equation theek hai. Lekin agar koi term mismatch kare (jaise $s=ut+\frac12 at$ mein $at$ ka dimension velocity ban jata hai), toh equation pakka galat hai. Yaad rakho: fail hona matlab definitely wrong, par pass hona matlab "shayad sahi" — kyunki dimensional analysis ko pure numbers (jaise $\frac12$ , $2\pi$ ) dikhte hi nahi.

Doosra use, jo exam mein points dilata hai: relation derive karna. Maan lo pendulum ka period $T$ depend karta hai $\ell$ , $m$ , $g$ par. Likho $T = k\,\ell^a m^b g^c$ , dono side ki $M, L, T$ ki powers match karo, aur tumhe $a=\frac12$ , $b=0$ , $c=-\frac12$ mil jaata hai — yaani $T \propto \sqrt{\ell/g}$ , aur mass ka koi role nahi! Constant $2\pi$ yeh method nahi dega, woh experiment se aata hai.

Bas limitation samajh lo: numeric constants, plus/minus signs, aur multi-term formulas yeh method nahi de sakta. Toh ise ek fast "sanity check" aur "first guess" tool ki tarah use karo — galti pakadne mein aur quick derivation mein yeh superpower hai.

Dimensional analysis — checking equations, deriving relations

WHAT are dimensions?

The Principle of Homogeneity (the engine)

USE 1 — Checking an equation

USE 2 — Deriving a relation (the powerful trick)

Limitations (know the boundaries)

80/20 — the few things that earn the marks

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Measurement, Vectors & Kinematics

Connections