Q.Convert the following into basic units:
🔒You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Unit Conversion
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: …
Concept: Unit Conversion using powers of ten.
The basic SI units here are metre (m) for length and gram (g) for mass. We use the prefix definitions: pico (p) = 10−12 and milli (m) = 10−3.
(i) 28.7 pm=28.7×10−12 m=2.87×10−11 m
(ii) 15.15 pm=15.15×10−12 m=1.515×10−11 m …
Unit conversion requires multiplying by the appropriate power of ten that relates the prefix to the base unit. (i) 28.7 pm=28.7×10−12 m,
(ii) 15.15 pm=15.15×10−12 m,
(iii) 25365 mg=2.5365×10−2 kg (the SI base unit of mass is the kilogram; that is 25.365 g).
The heart of unit conversion lies in understanding that prefixes are simply shorthand for powers of ten. When we write "pm" (picometre) or "mg" (milligram), we're using a compact notation for a number that would otherwise require many zeros. Converting to basic units means stripping away this prefix and expressing the quantity in metres, grams, or kilograms with the appropriate power of ten made explicit.
The SI system defines each prefix precisely. The prefix "pico-" means 10−12, so one picometre is one trillionth of a metre. The prefix "milli-" means 10−3, so one milligram is one thousandth of a gram. To convert, we replace the prefix with its numerical value.
(i) Converting 28.7 pm
-
Identify the prefix and base unit. We have "pm" which is picometre: the prefix is "pico-" and the base unit is metre (m).
-
Recall the prefix value. The prefix "pico-" represents 10−12.
-
Substitute and multiply. Replace "p" with 10−12:
28.7 pm=28.7×10−12 m
This is the quantity expressed in basic SI units (metres).
(ii) Converting 15.15 pm
-
Same prefix, same process. Again we have picometres, so "pico-" = 10−12.
-
Direct substitution:
15.15 pm=15.15×10−12 m
(iii) Converting 25365 mg
-
Identify the prefix and base unit. We have "mg" which is milligram: the prefix is "milli-" and the base unit is gram (g).
-
Recall the prefix value. The prefix "milli-" represents 10−3.
-
Substitute and multiply. Replace "m" with 10−3:
25365 mg=25365×10−3 g=25.365 g …
Method: Dimensional Analysis (Factor-Label Method)
This method uses conversion factors (ratios equal to 1) to cancel unwanted units and introduce desired ones.
Steps:
- Identify the given quantity and its unit.
- Write the target unit (basic SI unit).
- Choose the correct conversion factor (e.g., 1 pm=10−12 m, 1 mg=10−6 kg).
- Multiply the given value by the conversion factor so that the unwanted unit cancels.
- Simplify to get the answer in basic units.
(i) 28.7 pm → metres
- Given: 28.7 pm
- Target: metres (m)
- Conversion: 1 pm=10−12 m
28.7 pm×1 pm10−12 m=28.7×10−12 m
Answer: 2.87×10−11 m
(ii) 15.15 pm → metres
- Given: 15.15 pm
- Target: metres (m)
- Conversion: 1 pm=10−12 m
15.15 pm×1 pm10−12 m=15.15×10−12 m
Answer: 1.515×10−11 m
(iii) 25365 mg → kilograms
- Given: 25365 mg …
🧠 The Core Concept First
Basic units in the SI system are:
- Length: metre (m)
- Mass: kilogram (kg)
- Time: second (s)
So when asked to convert into "basic units", you must express the quantity in m (for length) or kg (for mass).
✗ Common Mistake #1: Forgetting the Prefix Meaning
Example: Treating pm (picometre) as 10⁻¹² m but then writing the answer incorrectly.
Why it happens: Students memorise prefixes but don't apply the exponent correctly — e.g., writing 28.7 pm = 28.7 × 10⁻¹² m instead of 2.87 × 10⁻¹¹ m.
✓ How to avoid:
- Write the conversion step explicitly:
1 pm=10−12 m
- Then multiply:
28.7 pm=28.7×10−12 m
- Always convert to scientific notation:
=2.87×10−11 m
✗ Common Mistake #2: Confusing pm with nm or μm
Example: Writing 15.15 pm = 15.15 × 10⁻⁹ m (that's nanometres, not picometres).
Why it happens: Prefixes like pico (p), nano (n), micro (μ) look similar under time pressure.
✓ How to avoid:
- Memorise the prefix ladder in order:
10−3 (milli), 10−6 (micro), 10−9 (nano), 10−12 (pico)
- Use a mnemonic: "My Nose Pops" → milli, nano, pico.
- Double-check: pico = p = 10⁻¹² (think "p" for "pint-sized").
✗ Common Mistake #3: Not Converting Mass to kg (the basic unit)
Example: Writing 25365 mg = 25365 × 10⁻³ g and stopping there.
Why it happens: Students think "gram" is the basic unit for mass — but the SI basic unit is kilogram (kg).
✓ How to avoid:
- Remember: kg is the base unit, not g.
- Two-step conversion:
1 mg=10−3 g
1 g=10−3 kg
So:
1 mg=10−6 kg
- Then:
25365 mg=25365×10−6 kg=2.5365×10−2 kg
✗ Common Mistake #4: Misplacing the Decimal in Scientific Notation
Example: Writing 28.7 pm = 2.87 × 10⁻¹⁰ m (off by one power of 10).
Why it happens: When converting 28.7 × 10⁻¹² to scientific notation, students shift the decimal incorrectly.
✓ How to avoid:
- Rule: In scientific notation, the coefficient must be between 1 and 10.
- For
28.7 × 10⁻¹²:- Move decimal one place left →
2.87 - Increase exponent by 1 →
10⁻¹¹
- Move decimal one place left →
- So:
28.7×10−12=2.87×10−11
--- …
Showing the 12 most recent of 17 on this concept.
- GUJCET 2026Set x1 markMCQQ.Out of the following physical quantities which quantity has the same unit as that of Planck's constant? (A) moment of force (B) power (C) angular momentum (D) moment of inertia
›Reveal solutionSolution
[h]=J⋅s, identical to the unit of angular momentum.
Planck's constant appears in E=hν, so [h]=[ν][E]=s−1J=J⋅s=kg⋅m2s−1.
Angular momentum L=mvr has units kg⋅(m/s)⋅m=kg⋅m2s−1=J⋅s. …
- GUJCET 2025Set 031 markMCQQ.The dimensional formula of current sensitivity of moving coil galvanometer is (A) [L2] (B) [M1L2T−2A−1] (C) [A−1] (D) [M1L2T−2]
›Reveal solutionSolution
[!TLDR]
Current sensitivity =θ/I; since θ is dimensionless, its dimension is [A−1].
Concept
The current sensitivity of a moving-coil galvanometer is the deflection produced per unit current, SI=Iθ=kNBA. An angular deflection is dimensionless.
Solution …
- GUJCET 2024Set 131 markMCQQ.Js is the unit of ________ physical quantity. (A) Angular momentum (B) Work function (C) Moment of Inertia (D) Rydberg constant
›Reveal solutionSolution
Angular momentum L=Iω has SI unit kg m2s−1=J⋅s. …
- GUJCET 2023Set 091 markMCQQ.Unit of mobility in terms of fundamental units is ______. (A) kg−1s−2A (B) kgs2A (C) kg−1s2A (D) kg−1s2A−1
›Reveal solutionSolution
[!TLDR]
Reducing m2V−1s−1 to fundamental units gives kg−1s2A.
Concept
Mobility μ=Evd has units of V/mm/s=m2V−1s−1. Convert the volt to base units to express μ in fundamental units.
Solution
V=CJ=Askgm2s−2=kgm2A−1s−3.
Then …
- GUJCET 2023Set 091 markMCQQ.The dimensional formula of self inductance is ______. (A) M1L1T−2A−2 (B) M1L2T−2A−2 (C) M−1L−1T2A2 (D) M1L−1T−1A−2
›Reveal solutionSolution
[!TLDR] [L]=M1L2T−2A−2 → (B).
Concept
Self-inductance is defined by ε=−LdtdI, so L=dIεdt. (NCERT Electromagnetic Induction.)
Solution …
- GUJCET 2023Set 091 markMCQQ.The earth takes 24 h to rotate once about its axis. How much time does the Sun takes to shift by 1 minute viewed from the earth. (A) 4 minutes (B) 40 s (C) 4 s (D) 40 minutes
›Reveal solutionSolution
[!TLDR]
The Sun moves 1° in 4 minutes, so 1 arcminute of apparent shift takes 4 seconds.
Concept
Earth's rotation makes the Sun appear to sweep 360∘ in 24 hours; a small angular shift corresponds to a proportional time.
Solution
Angular rate of the Sun's apparent motion:
24 h360∘=15∘/h=4 min1∘ …
- GSEB Higher Secondary Certificate (HSC) Examination 2023Set ANNUAL1 markMCQQ.The dimensional formula of electric flux is ___.(a) M^1 L^-3 T^-3 A^-1(b) M^1 L^3 T^-3 A^-1(c) M^-1 L^3 T^-3 A^-1(d) M^1 L^3 T^3 A^-1
›Reveal solutionSolution
Flux = E x area, i.e. (V/m) x m^2 = volt-metre; expressing volt in base units gives M L^3 T^-3 A^-1.
Electric flux Phi = E x A, with units (V/m)(m^2) = V.m.
…
- GUJCET 2021Set 151 markMCQQ.Which of the following option gives the Dimensional Formula of Electrical Potential? (A) [M−1L2T−3A1] (B) [M0L3T3A−1] (C) [M−1L−2T−4A2] (D) [M1L2T−3A−1]
›Reveal solutionSolution
Electric potential is energy per unit charge.
Concept: …
- GUJCET 2020Set 071 markMCQQ.The earth rotates on its axis takes 24 hours to complete one revolution. How much time it takes at sun from earth to have shift of 1∘? (A) 4 sec. (B) 4 hrs. (C) 4 min. (D) 24 hrs.
›Reveal solutionSolution
24 hours per 360∘ gives 4 minutes per degree.
Concept: Earth turns 360∘ in 24 h, so the angular rate is 360∘/24h=15∘/h. …
- GUJCET 2020Set 071 markMCQQ.Which is not the unit of Inductance? (A) H (B) V⋅s⋅A−1 (C) WbA−1 (D) Wb⋅s⋅A−1
›Reveal solutionSolution
1H=Wb/A=V⋅s/A; Wb⋅s⋅A−1 has an extra second and is not inductance. …
- GUJCET 2019Set 131 markMCQQ.The dimensional formula of effective torsional constant of spring is.......... (A) M0L0T0 (B) M1L2T−2 (C) M1L2T−2A−2 (D) M1L2T−3
›Reveal solutionSolution
Torsional constant has the dimensions of torque, M1L2T−2.
Concept: For a torsion spring, restoring torque τ=Cθ, so C=τ/θ. Angle θ is dimensionless, hence C has the dimensions of torque (energy).
Steps: …
- GUJCET 2019Set 131 markMCQQ.The dimensional formula of JWL is ................. Take Q as the dimension of charge. (A) M1L2T1Q−2 (B) M1L2T−1Q−2 (C) M1L−2T−1Q−2 (D) M−1L2T−1Q−2
›Reveal solutionSolution
The dimensional formula asked for is that of resistance (equivalently inductive reactance ωL): M1L2T−1Q−2.
Concept: Resistance R=IV. Expressing potential and current in terms of charge Q: V=Qenergy=QML2T−2 and I=TQ.
Steps:
- [R]=[I][V]=QT−1ML2T−2Q−1. …
🎓Unlock everything free for 14 days
- ✓Full step-by-step solutions
- ✓Concept-first explanations
- ✓Methods, shortcuts & mistakes
- ✓PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.