Q.Why do we have different units for the same physical quantity?
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Unit Conversion: Why 1 Metre and 100 Centimetres Are the Same Thing
Imagine you're measuring the length of your desk. You pull out a ruler marked in centimetres and find it's 120 cm long. Your friend, using a metre stick, says it's 1.2 m. You're both right — you've just used different units to describe the same physical length.
That's the core idea: unit conversion is the process of changing how you express a quantity without changing the quantity itself.
The Intuition: Same Quantity, Different Labels
Think of a pizza. Whether you call it "one pizza" or "8 slices," the amount of pizza hasn't changed. You've just used a different unit (pizza vs. slice) to describe it.
Similarly, 1 metre and 100 centimetres are the same length — just like 1 pizza and 8 slices are the same amount. The number changes (1 becomes 100, or 1 becomes 8), but the actual thing being measured stays identical.
This is the most important idea to hold onto: conversion changes the number, not the quantity. If you ever feel like the quantity has changed, you've made a mistake.
The Precise Statement
Unit conversion is the multiplication of a quantity by a conversion factor — a fraction equal to 1 — that cancels the old unit and introduces the new one.
A conversion factor looks like this:
old unitnew unit=1
For length: 100 cm1 m=1 and 1 m100 cm=1.
Why are these fractions equal to 1? Because 1 metre is 100 centimetres. The numerator and denominator describe the same physical length, so their ratio is exactly 1.
How to Convert: The Only Rule You Need
Multiply by a conversion factor that cancels the unit you have and leaves the unit you want.
Let's convert 120 cm to metres:
- Start with what you have: 120 cm
- Choose the conversion factor that has "cm" in the denominator (to cancel it) and "m" in the numerator: 100 cm1 m
- Multiply:
120 cm×100 cm1 m=100120 m=1.2 m
The "cm" units cancel just like numbers do: cmcm=1.
Always write the units explicitly. If the units don't cancel correctly, you've used the wrong conversion factor. This catches 90% of conversion mistakes.
The Reverse: Metres to Centimetres
Now convert 1.2 m to cm. This time, you want "cm" to remain and "m" to cancel. Use 1 m100 cm:
1.2 m×1 m100 cm=1.2×100 cm=120 cm
Notice: when going from a larger unit (m) to a smaller unit (cm), the number gets larger (1.2 → 120). When going from smaller to larger, the number gets smaller (120 → 1.2). This is a useful sanity check.
Common Conversion Factors You'll Use
| Quantity | Relationship | Conversion Factors |
|---|---|---|
| Length | 1 m = 100 cm | 100 cm1 m, 1 m100 cm |
| Mass | 1 kg = 1000 g | 1000 g1 kg, 1 kg1000 g |
| Time | 1 h = 60 min | 60 min1 h, 1 h60 min |
| Speed | 1 km/h = 36001000 m/s | 1 km1000 m×3600 s1 h |
Why this formula?
Dimensional Analysis: Why the Key Principles Hold
Dimensional Analysis is a powerful tool in physics and engineering that lets us check the consistency of equations, derive relationships, and convert units. But why does it work? Let's build the reasoning from the ground up.
1. The Core Idea: Physical Quantities Have Dimensions
Every physical quantity (like length, time, mass) can be expressed in terms of fundamental dimensions. The most common set in mechanics is:
- L = Length
- M = Mass
- T = Time
For example:
- Speed has dimensions [LT−1]
- Force has dimensions [MLT−2]
- Energy has dimensions [ML2T−2]
Why this matters: Two quantities can only be meaningfully compared or equated if they have the same dimensions. You cannot add apples to oranges — and you cannot add length to time.
2. The Principle of Dimensional Homogeneity
The key formula that underpins everything is:
Every valid physical equation must be dimensionally homogeneous.
This means: the dimensions on the left-hand side must equal the dimensions on the right-hand side.
Why must this hold?
Consider an equation like:
v=u+at
- Left side: [v]=LT−1
- Right side: [u]=LT−1, [at]=(LT−2)(T)=LT−1
Both sides have dimensions LT−1. If they didn't match, the equation would be physically meaningless — you'd be comparing quantities that cannot be equal in any real experiment.
Reasoning: Physical laws describe relationships between measurable quantities. If the dimensions don't match, the equation cannot represent a real physical relationship, because the numerical value would depend on the arbitrary choice of units.
3. The Buckingham Pi Theorem: Why We Can Derive Relationships
This is the deeper mathematical reason. The Buckingham Pi Theorem states:
If a physical problem involves n variables and k fundamental dimensions, then it can be reduced to n−k independent dimensionless groups (called π groups).
Why does this work?
Imagine you have a relationship:
f(Q1,Q2,…,Qn)=0
where each Qi has dimensions. Because the equation must be dimensionally homogeneous, we can rearrange it into a function of dimensionless products only:
F(π1,π2,…,πn−k)=0
The reasoning: Dimensions act as constraints. Each fundamental dimension (M, L, T) gives one constraint. So if you have n variables and k constraints, you only have n−k independent dimensionless combinations.
Example: For a simple pendulum, the period T depends on length L, mass m, and gravity g. That's 4 variables with 3 dimensions (M, L, T). So 4−3=1 dimensionless group: π=LT2g. This tells us T∝L/g without solving any differential equation.
4. Why We Can Convert Units Using Dimensional Analysis
The conversion factor formula:
Value in new unit=Value in old unit×(new unitold unit)dimension exponent
Why this works: …
Different measurement systems evolved independently in different regions before international standardisation, and even within one system, units of different sizes are chosen to keep numbers convenient for the scale being measured. …
The same physical quantity can be measured in different units because measurement systems developed independently in different places, and because a unit of convenient size is chosen for the scale being measured.
A physical quantity such as length, mass or time can be expressed using more than one unit for two main reasons:
-
Historical/regional development of unit systems. Measurement systems were developed independently by different civilisations and countries before international standardisation — e.g. the CGS system (centimetre-gram-second), the FPS/British system (foot-pound-second), and the MKS/SI system (metre-kilogram-second). Different countries adopted different systems for everyday use (e.g. miles/pounds in some countries, kilometres/kilograms in others), so the same quantity ends up with multiple valid units.
-
Convenience of scale. Even within one coherent system, sub-units and multiples exist so the numerical value of a measurement stays convenient for the context being described:
- Very small lengths are measured in nanometres or angstroms (atomic sizes), ordinary lengths in metres, and astronomical distances in light-years or parsecs — using the base SI unit (metre) for all of these would produce inconveniently tiny or huge numbers. …
Showing the 12 most recent of 37 on this concept.
- CBSE 2026Set ANNUAL1 markQ.1 Joule is equal to how many ergs?
›Reveal solutionSolution
1 J = 10⁷ erg.
The joule (SI) and erg (CGS) are both units of energy/work, defined as force × distance. 1 J = 1 kg·m²·s⁻², while 1 erg = 1 g·cm²·s⁻². Converting: 1 kg = 10³ g and 1 m = 10² cm, so 1 J = 1 kg …
- CBSE 2026Set ANNUAL1 markMCQQ.In SI system:(a) All derived units are obtained by multiplying (or) dividing the fundamental units.(b) All derived units are obtained by adding the fundamental units.(c) All derived units are obtained by subtracting the fundamental units.(d) Depends on the physical quantity.
›Reveal solutionSolution
Derived SI units always come from multiplying/dividing base units, never from adding them.
Every physical quantity's dimensional formula is built by raising the base dimensions (mass M, length L, time T, ...) to powers and combining them by multiplication/division. For example: velocity =L/T, force =MLT−2, energy =ML2T−2. Addition or subtraction is only defined between quantities of the same dimension (you cannot add a length to a time), so it can never be the rule used to build a new unit from the base units. The general construction rule for the whole SI system of derived units is therefore multiplication/division of the fundament …
- CBSE 2026Set ANNUAL1 markMCQQ.The submultiple 10^-2 has the prefix :(a) Centi(b) Hecto(c) Tera(d) Zepto
›Reveal solutionSolution
The prefix for the submultiple 10^-2 is centi.
SI prefixes are used to express very large or very small quantities as multiples or submultiples of a base unit, each prefix corresponding to a specific power of ten. Centi (symbol c) corresponds to 10^-2, hecto (h) corresponds to 10^2, tera (T) corresponds to 10^12, and zepto (z) corresponds to 10^-21. A familiar exa …
- CBSE 2026Set ANNUAL1 markQ.Write the answer in one word/one sentence: Write the value of 108 km/hour in m/s.
›Reveal solutionSolution
108 km/h converts to 30 m/s using the standard factor 5/18.
1 km/h = 1000 m / 3600 s = (5/18) m/s
…
- CBSE 2026Set ANNUAL1 markMCQQ.Unit of time is:(a) second(b) ampere(c) kelvin(d) meter
›Reveal solutionSolution
The SI base unit of time is the second (s).
The International System of Units (SI) defines seven base quantities, each with its own base unit. Ampere is the unit of electric current, kelvin is the unit of thermodynamic temperature, and metre is the unit of length. None of these measure time.
…
- CBSE 2026Set ANNUAL1 markMCQQ.Which of the following is a derived unit ?(a) ampere(b) mole(c) kelvin(d) joule
›Reveal solutionSolution
Ampere, mole and kelvin are base units; joule is derived. Answer (D).
The seven SI base units are: metre, kilogram, second, ampere, kelvin, mole and candela.
- ampere, mole, kelvin -> base units. …
- CBSE 2025Set ANNUAL1 markMCQQ.SI unit of energy joule is equivalent to (A) 10^6 erg (B) 10^-7 erg (C) 10^7 erg (D) 10^5 erg
›Reveal solutionSolution
1 joule equals 10^7 erg, found by converting mass and length between SI (kg, m) and CGS (g, cm) units.
Energy has dimensional formula [ML2T−2]. In SI, mass is measured in kg and length in m; in CGS, mass is in g and length in cm.
Conversion factors:
1 kg=103 g
1 m=102 cm⇒1 m2=104 cm2
So: …
- CBSE 2025Set ANNUAL1 markMCQQ.How many scientific fundamental quantities are given in SI units?(a) 5(b) 7(c) 3(d) 9
›Reveal solutionSolution
The SI system recognises 7 base physical quantities; every other unit (like N, J, Pa, C) is derived from these.
The International System of Units (SI) is built on a small set of base quantities that are chosen to be mutually independent — no one can be expressed in terms of the others. NCERT Class 11 Chemistry (Unit 1) lists these seven base quantities and their SI units:
- Length — metre (m)
- Mass — kilogram (kg)
- Time — second (s)
- Electric current — ampere (A)
- Thermodynamic temperature — kelvin (K)
- Amount of substance — mole (mol)
- Luminous intensity — candela (cd) …
- CBSE 2025Set hz1 markMCQQ.The correct relation between Light year and metre is:(a) 1 Light year = 7.469 x 10^15 m(b) 1 Light year = 4.2 m(c) 1 Light year = 9.467 x 10^15 m(d) None of them
›Reveal solutionSolution
1 light year = speed of light x time in 1 year approx 9.467 x 10^15 m.
A light year is defined as the distance travelled by light in vacuum in one year. To compute it:
Speed of light, c = 3 x 10^8 m/s
1 year = 365.25 days x 24 hours x 3600 seconds approx 3.156 x 10^7 s
Distance = c x t = (3 x 10^8 m/s) x (3.156 x 10^7 s) approx 9.467 x 10^15 m
…
- CBSE 2025Set ANNUAL1 markQ.Fill in the blank: The volume of a cube of side 2 cm is equal to ......... m^3.
›Reveal solutionSolution
The volume of a 2 cm side cube is 8 cm^3, which equals 8 x 10^-6 m^3.
Side of cube, a = 2 cm.
Volume, V = a^3 = (2 cm)^3 = 8 cm^3. …
- CBSE 2025Set ANNUAL1 markQ.Fill in the blank: G = 6.67 x 10^-11 Nm^2 kg^-2 = ......... cm^3 s^-2 g^-1.
›Reveal solutionSolution
Converting G = 6.67 x 10^-11 Nm^2 kg^-2 into cgs units gives 6.67 x 10^-8 cm^3 s^-2 g^-1.
First express G in base SI units: since N = kg m s^-2, Nm^2 kg^-2 = (kg m s^-2)(m^2) kg^-2 = m^3 kg^-1 s^-2.
So G = 6.67 x 10^-11 m^3 kg^-1 s^-2.
Now convert m to cm: 1 m^3 = (10^2 cm)^3 = 10^6 cm^3.
Convert kg^-1 to g^-1: 1 kg^-1 = (10^-3)^-1 g^-1... more directly, since 1 kg = 10^3 g, 1 kg^-1 = 10^-3 g^-1. …
- CBSE 2025Set ANNUAL1 markQ.Answer in one word or one sentence: How many metre is equivalent to 1 Fermi?
›Reveal solutionSolution
1 Fermi equals 10^-15 metre, the typical scale of atomic nuclei.
The fermi (symbol fm), also called the femtometre, is a unit of length used to express nuclear and sub-nuclear distances, since nuclear radii are of …
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