Dimensions and Dimensional Formulae

RPF Constable reasoning papers love the basic syllogism pair: two statements of the "All A are B" form, two conclusions to test. The right way to handle them is not by gut feeling — it is by drawing one tiny Venn diagram and applying one chain rule. Once you see the pattern below, this kind of question becomes free marks.

Definition: A syllogism is a form of logical reasoning where two or more statements (called premises) are given, and we must decide which conclusion (or conclusions) necessarily follow.

Definition: A statement of the form "All A are B" is called a universal affirmative (Type A in classical logic). On a Venn diagram, it means the entire circle of A lies inside the circle of B.

The problem in front of us

Statements

1. All pens are books.
2. All books are tables.

Conclusions

• I. All pens are tables.
• II. Some tables are pens.

We must judge whether each conclusion necessarily follows from the statements.

Step 1 — Draw the Venn diagram

"All pens are books" puts the pens circle entirely inside the books circle.
"All books are tables" puts the books circle entirely inside the tables circle.

So we get three nested circles, from inside out: pens ⊂ books ⊂ tables.

Step 2 — Check Conclusion I: "All pens are tables"

Take any pen. It lies inside the books circle (by statement 1). The books circle lies inside the tables circle (by statement 2). So that pen lies inside the tables circle. Since this is true for every pen, every pen is a table — Conclusion I follows.

This is the chain rule of categorical logic:
All A are B + All B are C ⇒ All A are C.

Step 3 — Check Conclusion II: "Some tables are pens"

"Some" means at least one. From "All pens are tables," every single pen is a table. The instant at least one pen exists, at least one table is a pen — which is exactly what Conclusion II says. So Conclusion II also follows.

This rule has a name in classical logic: conversion by limitation of a universal affirmative — from "All A are B," we may infer "Some B are A" (provided A is non-empty, which is the standard assumption in exam syllogism).

Step 4 — Combine the conclusions

Both Conclusion I and Conclusion II follow. The correct answer choice on the RPF format is therefore "Both I and II follow."

Why the chain rule is the most-tested pattern

In RPF, SSC and Railways reasoning, "All A are B + All B are C" appears in roughly one out of every three syllogism questions. The reason is that it cleanly tests transitivity — the same property that makes mathematics work. Examiners can dress it up with weird nouns (pens, books, tables, mangoes, dancers, painters), but the underlying logic is unchanged.

Equally, the immediate inference "All A are B ⇒ Some B are A" (i.e. converting "All" to "Some") catches students who only check the headline conclusion and miss the easier one hiding underneath.

Why it matters: A correctly solved syllogism is one of the highest-confidence marks in the entire RPF reasoning paper. There is no ambiguity, no opinion — if you draw the Venn diagram correctly, the answer is forced. Build a 10-second routine: read statements → draw circles → trace each conclusion → tick which follow.

Real-world example: The same logic appears in real-life claims. "All RPF constables wear the prescribed uniform; all members of the uniform brigade salute the National Flag" gives you instantly: all RPF constables salute the National Flag (chain rule), and at least one of those who salute the National Flag is an RPF constable (conversion). Syllogism is just spotting this pattern under exam pressure.

Common misconception: Students sometimes accept Conclusion I but reject Conclusion II, thinking "Some tables are pens" is weaker and therefore not allowed if "All pens are tables" is already true. The opposite is true: "All" is stronger than "Some," and a stronger truth implies the weaker one. Both follow.

Another mix-up: trying to apply the chain rule to mixed statements. "All A are B + Some B are C" does NOT give "Some A are C," because the "some" B that are C might be precisely the part of B outside A. Always check: the middle term ("B" here) must be universally distributed to chain.

Question: Given Statements (1) All pens are books, (2) All books are tables, and Conclusions (I) All pens are tables, (II) Some tables are pens — which conclusions follow?
Solution:
Step 1: Draw the Venn — pens inside books inside tables.
Step 2: Trace any pen: it sits inside books, hence inside tables ⇒ All pens are tables ⇒ Conclusion I follows.
Step 3: Since every pen is a table, at least one table (each pen) is a pen ⇒ Some tables are pens ⇒ Conclusion II follows.
Step 4: Both follow.
Conclusion: The correct answer is Both I and II follow — by the chain rule (for I) and conversion of "All" to "Some" (for II).

Uses: (1) checking dimensional correctness of equations (principle of homogeneity - all terms must have same dimensions); (2) deriving relations among quantities; (3) converting units from one system to another. Limitations: (1) cannot determine dimensionless constants (like 1/2, pi, 2); (2) cannot derive relations involving sum/difference of terms; (3) fails for trigonometric, exponential, logarithmic functions; (4) cannot work if a quantity depends on more than 3 factors with M, L, T. Key rule: arguments of sin, cos, log, e^x are always dimensionless. Quantities with same dimensions but different nature: work and torque; stress and pressure and Young's modulus.

To convert a quantity from one system to another: n1[M1^a L1^b T1^c] = n2[M2^a L2^b T2^c], so n2 = n1 (M1/M2)^a (L1/L2)^b (T1/T2)^c. Example: Convert 1 joule to erg. Joule = [ML^2T^-2], so a=1, b=2, c=-2. n2 = 1 x (kg/g)^1 (m/cm)^2 (s/s)^-2 = 1 x (1000)(100^2)(1) = 1000 x 10000 = 10^7. So 1 J = 10^7 erg. Always raise the ratio of OLD to NEW unit to the power of the dimension.

Dimensional analysis is the fastest way to catch a wrong formula in the exam hall and to reconstruct a forgotten relation. NEET almost always includes one MCQ that reduces to "which of these has the same dimensions as X". Learn the standard dimensional formulae cold and this becomes a free mark.

The core idea

The dimension of a quantity shows how it is built from mass [M], length [L] and time [T] (plus [A], [K] where needed). A dimensional formula like [M L² T⁻²] is a fingerprint. The principle of homogeneity says every additive term in a valid equation must share the same dimensions.

Must-know dimensional formulae

Quantity	Dimensional formula
Velocity	[M⁰ L T⁻¹]
Acceleration	[M⁰ L T⁻²]
Force	[M L T⁻²]
Work / Energy	[M L² T⁻²]
Power	[M L² T⁻³]
Momentum / Impulse	[M L T⁻¹]
Pressure / Stress / Young's modulus	[M L⁻¹ T⁻²]
Surface tension / Spring constant	[M L⁰ T⁻²]
Coefficient of viscosity	[M L⁻¹ T⁻¹]
Planck's constant / Angular momentum	[M L² T⁻¹]
Gravitational constant G	[M⁻¹ L³ T⁻²]

Uses and limits

Uses: check correctness (homogeneity), convert units between systems, derive a relation up to a dimensionless constant.
Limits: cannot find dimensionless constants (like ½ or 2π); fails when arguments are trig/exp/log; cannot derive equations with more than three unknowns; cannot distinguish quantities with identical dimensions.

Exam Tricks & Tips

• 🎯 Work, torque and energy all share [M L² T⁻²] — dimensions cannot tell them apart.
• 🎯 Planck's constant h has the same dimensions as angular momentum, [M L² T⁻¹].
• 🎯 Surface tension and spring/force constant share [M T⁻²].
• 🎯 Pressure, stress, energy density and Young's modulus all give [M L⁻¹ T⁻²].
• 🎯 Whatever multiplies a sine or sits in an exponent must be dimensionless — a fast way to find unknown powers.
• ❌ Homogeneity confirms dimensional consistency only; a formula can be dimensionally correct yet still wrong by a pure number.

Expected exam pattern

"Which pair has the same dimensions?", finding the dimensional formula of a constant (like a or b in van der Waals gas equation), or checking whether a given equation is dimensionally valid.

Quick recap

Reduce every quantity to [M^a L^b T^c]. Memorise the table above, use homogeneity to test equations, and remember the three big limitations. Same dimensions ≠ same quantity.

Cover the answer, recall, then check. 12 cards on dimensional formulae.

Q1. Dimensional formula of force.
A1. [M L T⁻²] (mass × acceleration).

Q2. Dimensional formula of work, energy and torque.
A2. All are [M L² T⁻²].

Q3. Dimensional formula of power.
A3. [M L² T⁻³] (energy per unit time).

Q4. Dimensions of pressure, stress and Young's modulus.
A4. All [M L⁻¹ T⁻²] (force per area).

Q5. Which quantity shares dimensions with Planck's constant?
A5. Angular momentum — both [M L² T⁻¹].

Q6. Dimensional formula of the gravitational constant G.
A6. [M⁻¹ L³ T⁻²], from G = Fr²/(m₁m₂).

Q7. Dimensions of coefficient of viscosity.
A7. [M L⁻¹ T⁻¹] (SI unit Pa·s).

Q8. Dimensions of surface tension and of spring constant.
A8. Both [M T⁻²] (force per unit length).

Q9. State the principle of homogeneity.
A9. Every additive term in a physically valid equation must have the same dimensions.

Q10. Give two things dimensional analysis cannot do.
A10. Find dimensionless constants (½, 2π) and handle trig/log/exp; it also can't derive relations with more than 3 unknowns.

Q11. Dimensions of momentum and of impulse.
A11. Both [M L T⁻¹] (impulse equals change in momentum).

Q12. The argument of sin, log or e^x must be…?
A12. Dimensionless — this constraint often fixes unknown powers in a formula.