PhysicsClass 11CBSE

How to Catch a Wrong Physics Formula Without Doing an Experiment

Write dimensional formulas in terms of M, L and T, test equations for dimensional consistency, derive relations such as the pendulum's time period by dimensional analysis, know its limits, and convert quantities between unit systems.

What can the dimensions of a quantity tell you?

Every mechanical quantity is built from just three base quantities: mass (M), length (L) and time (T). Speed is length divided by time; force is mass times length divided by time squared.

Writing quantities this way lets you check a formula, derive a new one, and convert units — often in a few lines and without any experiment.

This part covers dimensional formulas, dimensional consistency, deriving relations, and converting between systems of units.

How do you write the dimensional formula of a physical quantity?

**The dimensional formula shows the powers of M, L and T that make up a quantity, written as

[M^{a} L^{b} T^{c}]

; the dimensional equation sets the quantity equal to that formula.

Worked examples.

- Velocity**

= \frac{length}{time}

[M^{0} L T^{- 1}]

- Acceleration:

[M^{0} L T^{- 2}]

- Force

= ma

[M L T^{- 2}]

- Work

= F s

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

- Power

= \frac{W}{t}

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

- Pressure

= \frac{F}{A}

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

Worked example — a constant. From

F = \frac{G m _{1} m _{2}}{r ^{2}}

G = \frac{F r ^{2}}{m _{1} m _{2}}

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

The dimensional equation is

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

.

Dimensionless quantities such as angle, strain and refractive index have

[M^{0} L^{0} T^{0}]

.

An everyday example. The pressure inside a pressure cooker and the stress in a steel rod both have dimensions

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

, because both are force per area.

The substance. Same dimensions do not mean the same quantity — work and torque are both

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

How do you check whether an equation is dimensionally correct?

By the principle of homogeneity, every term added, subtracted or equated must have the same dimensions; if any term differs, the equation is certainly wrong.

Worked example 1 — a correct equation. For

v^{2} = u^{2} + 2 a s

[L T^{- 1}]^{2} = [L T^{- 1}]^{2} + [L T^{- 2}] [L] \Rightarrow [L^{2} T^{- 2}] = [L^{2} T^{- 2}] + [L^{2} T^{- 2}]

Every term matches, so it is dimensionally consistent.

Worked example 2 — spotting impossible formulas for kinetic energy.

-

K = m^{2} v^{3}

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

— impossible
-

K = ma

[M L T^{- 2}]

— impossible
-

K = \frac{1}{2} m v^{2}

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

— consistent
-

K = \frac{3}{16} m v^{2}

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

— consistent but wrong

Worked example 3 — inside a function. In

x = A sin ω t

, the angle

ω t

must be dimensionless, so

[ω] = [T^{- 1}]

.

An everyday example. You cannot add 5 kg of rice to ₹200 — the answer means nothing, just as adding a length to a time does.

The substance. Dimensional consistency is necessary but not sufficient — the last kinetic energy formula passes the test but has the wrong number.

How do you derive a relationship using dimensional analysis, and where does it fail?

Assume the quantity depends on the others as a product of powers with a dimensionless constant, equate the powers of M, L and T on both sides, and solve for the unknown powers.

Worked example 1 — simple pendulum. Suppose

T = k l^{a} m^{b} g^{c}

[T] = [L]^{a} [M]^{b} [L T^{- 2}]^{c}

- M:

b = 0

- T:

- 2 c = 1

, so

c = - \frac{1}{2}

- L:

a + c = 0

, so

a = \frac{1}{2}

T = k \frac{l}{g}

Theory gives

k = 2 π

, so a

1

m pendulum with

g = 9.8

m s

^{- 2}

has

T = 2 π \frac{1}{9.8} \approx 2.0

s.

Worked example 2 — wave on a string. With tension

F

[M L T^{- 2}]

and mass per length

μ

[M L^{- 1}]

v = k F^{a} μ^{b}

gives

a + b = 0

a - b = 1

- 2 a = - 1

, so

v = k \frac{F}{μ}

.

Limitations:

- Cannot find dimensionless constants such as

2 π

- Cannot derive equations with sums like

s = u t + \frac{1}{2} a t^{2}

- Fails for trigonometric, exponential and logarithmic relations
- Works only when unknowns do not exceed the three base dimensions

An everyday example. A park swing with a chain four times longer swings back and forth half as often, since

T \propto l

.

The substance. The mass drops out — the pendulum's period does not depend on the bob's mass.

How do you convert a quantity from one system of units to another using dimensions?

**Since the physical quantity is unchanged,

n_{1} u_{1} = n_{2} u_{2}

, so for dimensions

[M^{a} L^{b} T^{c}]

the new number is

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

.

Worked example 1 — joule to erg.** Work is

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

;

1

= 1 0^{3}

1

= 1 0^{2}

cm.

n_{2} = 1 \times (1 0^{3})^{1} \times (1 0^{2})^{2} \times 1 = 1 0^{7} \Rightarrow 1 J = 1 0^{7} erg

Worked example 2 — newton to dyne. Force is

[M L T^{- 2}]

n_{2} = 1 0^{3} \times 1 0^{2} = 1 0^{5}

, so

1

= 1 0^{5}

dyne.

Worked example 3 — the gravitational constant.

G = 6.67 \times 1 0^{- 11}

in SI, with

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

n_{2} = 6.67 \times 1 0^{- 11} \times (1 0^{3})^{- 1} \times (1 0^{2})^{3} = 6.67 \times 1 0^{- 8} in CGS

Worked example 4 — a made-up system. If the units of mass, length and time are

10

kg,

10

m and

10

s, then

100

J becomes

n_{2} = 100 \times (\frac{1}{10})^{1} \times (\frac{1}{10})^{2} \times (\frac{1}{10})^{- 2} = 10

An everyday example. Rice at ₹80 per kg is

8000

paise per

1000

g, which is

8

paise per gram — the same price in smaller units.

The substance. A bigger unit always gives a smaller number for the same quantity.

Exam tip

What earns full marks on dimensions?

Start every dimensional question by writing the defining formula of each quantity, then substitute known dimensions step by step.

- Memorise force

[M L T^{- 2}]

, energy

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

, power

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

, pressure

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

- Homogeneity: every added term has the same dimensions
- Arguments of sin, cos, log and exponentials are dimensionless
- Derivations: equate powers of M, L, T separately
- Conversion:

n_{2} = n_{1} [M_{1} / M_{2}]^{a} [L_{1} / L_{2}]^{b} [T_{1} / T_{2}]^{c}

The trap. Claiming a formula is correct because it is dimensionally consistent. Consistency can only prove a formula wrong, never right.

Did you know

How can dimensions alone predict the speed of ocean waves?

The speed

v

of waves on deep water can only depend on **gravity

g

and the wavelength

λ

.** Try

v = k g^{a} λ^{b}

[L T^{- 1}] = [L T^{- 2}]^{a} [L]^{b} \Rightarrow a = \frac{1}{2}, b = \frac{1}{2} \Rightarrow v = k g λ

Without any experiment, this shows that longer waves travel faster. The full theory gives

k = \frac{1}{2 π}

, so waves

100

m apart move at about

\frac{9.8 \times 100}{2 π} \approx 12.5

m s

^{- 1}

Exam relevance

How are dimensions tested in JEE Main and NEET?

Dimensional analysis belongs to Units and Measurements in both JEE Main and NEET, and JEE Advanced uses it inside harder problems.

What gets asked. Dimensions of less familiar quantities such as Planck's constant, permittivity or coefficient of viscosity; finding the dimensions of constants

a

and

b

in a given equation; checking which formula is possible; and match-the-column lists of quantities with their dimensions.

Question types. Mostly multiple-choice; assertion-reason statements about homogeneity also appear.

The trap that costs marks. Forgetting that the argument of sin or an exponential is dimensionless, which is often the key step for finding a constant's dimensions.

Key takeaways

What must you be able to do from this part?

- Dimensional formulas: force

[M L T^{- 2}]

, work

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

G

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

- Homogeneity:

v^{2} = u^{2} + 2 a s

is consistent;

K = m^{2} v^{3}

is impossible
- Derivation: pendulum

T = k l / g

; mass does not appear
- Limitations: no constants, no sums, no trig or exponential relations
- Conversion:

1

= 1 0^{7}

erg;

1

= 1 0^{5}

dyne

Use dimensional analysis to check whether

s = u t + \frac{1}{3} a t^{3}

could be correct, and explain what your answer proves.

What can the dimensions of a quantity tell you?

How do you write the dimensional formula of a physical quantity?

How do you check whether an equation is dimensionally correct?

How do you derive a relationship using dimensional analysis, and where does it fail?

How do you convert a quantity from one system of units to another using dimensions?

What earns full marks on dimensions?

How can dimensions alone predict the speed of ocean waves?

How are dimensions tested in JEE Main and NEET?

What must you be able to do from this part?

Related study notes

Ready to put this into practice?