NCERT Plus One (Class 11) Physics • Chapter 1

Units and Measurements

Q: Name two pairs of physical quantities having the same dimensional formula.

Key (1 Mark): (i) Work & Torque — both [M L² T⁻²] . (ii) Pressure & Stress (or Young's Modulus) — both [M L⁻¹ T⁻²] . (iii) Momentum & Impulse — both [M L T⁻¹] .

Q: Check the dimensional correctness of the equation ½mv² = mgh.

LHS: [½mv²] = [M][LT⁻¹]² = [ML²T⁻²] . RHS: [mgh] = [M][LT⁻²][L] = [ML²T⁻²] . Since [LHS] = [RHS] , the equation is dimensionally correct.

Q: Each side of a cube is measured to be 7.203 m. Find the total surface area and volume of the cube to appropriate significant figures.

7.203 m has 4 sig figs . • Surface Area = 6 × (7.203)² = 311.299... m² → 311.3 m² (4 sig figs; note 6 is exact). • Volume = (7.203)³ = 373.714... m³ → 373.7 m³ (4 sig figs).

Q: The centripetal force F acting on a particle moving uniformly in a circle depends upon its mass m, velocity v, and radius r. Derive an expression for F using dimensional analysis.

Let F = k mᵃ vᵇ rᶜ . [MLT⁻²] = [Mᵃ] [LT⁻¹]ᵇ [Lᶜ] = [Mᵃ Lᵇ⁺ᶜ T⁻ᵇ] . Equating exponents: a = 1 , −b = −2 ⇒ b = 2 , b + c = 1 ⇒ c = −1 . Hence F = k (mv² / r) (where experiment gives k = 1 ).

Q: In the relation y = A sin(ωt − kx), where y & x are distances and t is time, find the dimensional formulae of ω and k.

Angles inside sin(...) are dimensionless [M⁰L⁰T⁰] : • [ωt] = 1 ⇒ [ω] = [T⁻¹] • [kx] = 1 ⇒ [k] = [L⁻¹]

Q: State any two limitations of dimensional analysis.

1. It cannot determine dimensionless numerical constants like ½ or 2π . 2. It cannot derive equations involving multiple added/subtracted terms (like s = ut + ½at² ) or trigonometric/exponential/logarithmic functions.

Q: A calorie is about 4.2 J where 1 J = 1 kg m² s⁻². Suppose we adopt a new unit system with mass = α kg, length = β m, time = γ s. Express 1 calorie in the new units.

n₂ = n₁ (M₁/M₂)¹ (L₁/L₂)² (T₁/T₂)⁻² = 4.2 × (1 kg / α kg)¹ (1 m / β m)² (1 s / γ s)⁻² = 4.2 α⁻¹ β⁻² γ² new units.

Q: "A dimensionally correct equation need not be physically exact, whereas a dimensionally wrong equation must be wrong." Justify with one example.

Correct! K = mv² is dimensionally right ([ML²T⁻²]) yet physically wrong by factor ½. Conversely, F = mv/t² has mismatched dimensions ([MLT⁻³] ≠ [MLT⁻²]) so it can NEVER represent force.

SI base & supplementary units, significant figures, arithmetic rounding, dimensions & dimensional analysis.

📄 Official NCERT Textbook PDF

Topic 1.1 – 1.2: SI System & Base / Supplementary Units

Imagine ordering three "arm-lengths" of ribbon in Kozhikode and finding out your cousin's "arm-length" in Kochi gives a completely different piece! Before 1971, scientists had the exact same headache with inches, feet, and local pounds. That is why every country agreed on one shared ruler book — the SI System (International System of Units) — so an engineer in ISRO Thiruvananthapuram and a lab in Geneva read the exact same numbers.

7 SI BASE UNITSm (length), kg (mass), s (time), A (current), K (temp), mol (amount), cd (light)

2 ANGLE UNITSPlane angle = rad (s/r); Solid angle = sr (dA/r²)

DIMENSIONLESS BUT HAS UNITAngle (rad) & Solid angle (sr) have units, zero dimensions!

🧠 Memory Trick: Remember 7 base quantities with "Meera Loves Tea And Kattan Molten Candy" → Mass, Length, Time, Ampere, Kelvin, Mole, Candela!

💡 Simple Kid-Check: Measuring anything is just comparing an unknown thing against one fixed standard piece — like checking how many 1-metre sticks fit across your classroom floor!

1
Base Quantities (Independent 7): Cannot be built from other base units: Length (m), Mass (kg), Time (s), Electric current (A), Thermodynamic temperature (K), Amount of substance (mol), and Luminous intensity (cd).
2
Supplementary Units (Angles): Plane angle dθ = ds / r in radian (rad), where a full circle is 2π rad = 360°. Solid angle in 3D cone dΩ = dA / r² in steradian (sr), where a full sphere is 4π sr.
3
Derived Units: Everything else (speed m/s, force N = kg·m/s², pressure Pa) is built by multiplying or dividing the 7 base units.

Base Quantity	SI Unit	Symbol	Dim. Letter
Length	metre	m	[L]
Mass	kilogram	kg	[M]
Time	second	s	[T]
Electric Current	ampere	A	[A]
Temperature	kelvin	K	[K]
Amount of Substance	mole	mol	[mol]
Luminous Intensity	candela	cd	[cd]

Watch the classic Kerala HSE 1-mark trap: Plane angle (radian) and Solid angle (steradian) HAVE SI units, yet they are dimensionless [M⁰L⁰T⁰] because they are ratios of length/length and area/area!

Topic 1.3: Significant Figures & Rounding Off Rules

Suppose you weigh your school bag on a bathroom scale and read 4.2 kg, while a goldsmith weighs a gold ring and reads 4.200 g. Those extra trailing zeroes after the decimal point are NOT lazy decorations — they brag about how sharp and trustworthy the measuring instrument is!

ALWAYS COUNTAll non-zero digits (1–9) & trapped zeroes (e.g. 2004 → 4)

LEADING ZEROESNEVER significant! (0.0045 → only 2 sig figs)

TRAILING ZEROESWith decimal: YES (3.500 → 4). Without decimal: NO (4700 m → 2)

🧠 Memory Trick: When rounding off an exact 5 (with nothing/zeroes after), make the preceding digit EVEN: 2.745 → 2.74, 2.735 → 2.74!

💡 Simple Kid-Check: Significant figures count every digit you are sure about plus one last slightly uncertain digit. Changing units (like 4.700 m = 470.0 cm) must NEVER change the number of significant figures!

1
Rule 1 & 2: Every non-zero digit and any zero trapped between non-zero digits counts (1.008 has 4 sig figs).
2
Rule 3 (Small numbers < 1): Zeroes to the left of the first non-zero digit NEVER count (0.002308 has only 4 sig figs: 2, 3, 0, 8).
3
Rule 4 & 5 (Trailing zeroes): In 12300 cm (no decimal), trailing zeroes do NOT count (3 sig figs). With a decimal point like 3.500 or scientific notation 4.700 × 10², trailing zeroes DO count (4 sig figs)!
4
Rule of 5 (Rounding): Drop >5 → add 1; drop <5 → keep same. If dropped digit is exact 5: leave preceding digit unchanged if even, add 1 if odd!

Trap Alert: 4.700 m = 470.0 cm = 4.700 × 10³ mm — ALL have 4 significant figures! Never write 4700 mm if you measured to 4 sig figs; use scientific notation a × 10ᵇ so precision stays crystal clear.

Topic 1.3.1: Arithmetic Rules: Adding vs Multiplying Measurements

Suppose you weigh a watermelon on a big market scale as 3.2 kg and sprinkle 0.0042 kg of spice measured on a lab balance. Can you claim the total is 3.2042 kg? Nope! Your big market scale isn't even sure about the second decimal digit, so the final answer must bow down to the least precise measurement!

ADD / SUBTRACTKeep fewest DECIMAL PLACES (after decimal dot)!

MULTIPLY / DIVIDEKeep fewest total SIGNIFICANT FIGURES!

EXACT NUMBERSCounts like 2 in 2πr have infinite sig figs — never limit!

🧠 Memory Trick: A-D = Addition matches Decimal places! M-S = Multiplication matches Significant figures!

💡 Simple Kid-Check: A chain is only as strong as its weakest link! Think of decimal places for adding/subtracting, and total significant digits for multiplying/dividing.

🧒 Worked NCERT Comparison:
• Addition: 436.32 g + 227.2 g + 0.301 g = 663.821 g. Fewest decimal places = 1 (from 227.2) → Round to 663.8 g!
• Multiplication/Division: Density of 4.237 g in 2.51 cm³ = 1.68804.... Fewest sig figs = 3 (from 2.51) → Round to 1.69 g/cm³!

Do NOT apply the significant-figures rule to addition/subtraction or the decimal-place rule to multiplication! For example, 2.00 + 3.1234 = 5.12 (2 decimal places, NOT 3 sig figs — well, here 5.12 has 3 sig figs anyway, whereas 100.1 + 0.25 = 100.4 keeps 1 decimal place = 4 sig figs!).

Topic 1.4: Dimensions of Physical Quantities

Can you trade 5 metres of cloth directly for 5 seconds of music? No — they belong to totally different families! Dimensions strip off units like km, cm, grams, or hours and reveal the secret family DNA inside any quantity: how many copies of Mass [M], Length [L], and Time [T] built it!

WHAT ARE DIMENSIONS?Powers (a, b, c) raised to base quantities [Mᵃ Lᵇ Tᶜ]

BIG 4 IN MECHANICSForce [MLT⁻²] • Work/Energy [ML²T⁻²] • Power [ML²T⁻³] • Pressure [ML⁻¹T⁻²]

FAMOUS TWINSWork & Torque [ML²T⁻²]; Pressure, Stress & Young's Modulus [ML⁻¹T⁻²]; Impulse & Momentum [MLT⁻¹]

🧠 Memory Trick: Start with Speed [LT⁻¹] → multiply by T⁻¹ for Acceleration [LT⁻²] → multiply by M for Force [MLT⁻²] → multiply by L for Work [ML²T⁻²]!

💡 Simple Kid-Check: Speed uses 1 Length divided by 1 Time, whether you measure it in km/hour or cm/second. So speed's dimension is ALWAYS [M⁰ L¹ T⁻¹] — dimensions care about type of quantity, not unit size!

Volume = [L³] | Density = [ML⁻³] | Velocity = [LT⁻¹] | Momentum = [MLT⁻¹]
Force = Mass × Accel = [M¹L¹T⁻²] | Universal Gravitation G = [M⁻¹L³T⁻²]

Watch out: Quantities with different physical meanings can share identical dimensions — like Torque (rotational turning effect) and Work (scalar dot product), both [ML²T⁻²]! So same dimensions do NOT prove two quantities are the same concept.

Topic 1.5: Dimensional Formulae & Principle of Homogeneity

Could you ever solve 5 apples + 2 bicycles = ? Of course not — apples and bicycles cannot be added! In Physics, Nature obeys that exact rule on steroids: every single term separated by +, −, or = in a valid equation MUST have the exact same dimensional DNA!

HOMOGENEITY RULELHS = RHS AND every added/subtracted term has identical dimensions!

DIMENSIONLESS TRAPSNumbers (2, π), angles, ratios, trig/log/exp arguments must be [M⁰L⁰T⁰]

CONSTANT SOLVINGIn v = A + Bt: [A] = [v] = LT⁻¹, [Bt] = LT⁻¹ ⇒ [B] = LT⁻²

🧠 Memory Trick: Think "Same Shelf Rule" — you can only add [LT⁻¹] + [LT⁻¹] on the same shelf, never [L] + [LT⁻¹]!

💡 Simple Kid-Check: Whenever an equation has X = Y + Z, immediately set [X] = [Y] = [Z]! That instantly unlocks mystery constants like van der Waals a and b or refractive index coefficients!

Quick Check Example: In s = ut + ½ at²:
[s] = [L]; [ut] = [LT⁻¹][T] = [L]; [½ at²] = [LT⁻²][T²] = [L] → All match [L]!

Mystery Constants Example: In (P + a/V²)(V − b) = RT: since a/V² adds to pressure P, [a] = [P][V²] = [ML⁻¹T⁻²][L⁶] = [ML⁵T⁻²], and [b] = [V] = [L³]!

If sin(ωt − kx) appears inside a wave equation, remember the whole angle (ωt − kx) MUST be dimensionless! That instantly gives [ω] = [T⁻¹] and [k] = [L⁻¹].

Topic 1.6: Deducing New Equations & Limitations of Dimensional Analysis

Suppose you land on an alien planet and forget the pendulum clock formula! Using ONLY dimensional analysis — testing how length l, mass m, and gravity g combine to produce seconds [T] — you magically rediscover T = k √(l/g) without doing a single calculus step!

4-STEP RECIPEWrite Q ∝ XᵃYᵇZᶜ → equate powers of M, L, T → solve simultaneous equations!

UNIT CONVERSIONn₂ = n₁ (M₁/M₂)ᵃ (L₁/L₂)ᵇ (T₁/T₂)ᶜ (e.g. 1 N = 10⁵ dyne)

3 LIMITATIONSCan't find pure number k (e.g. 2π); fails for + / − equations or sin/exp/log; max 3 unknowns!

🧠 Memory Trick: A dimensionally correct equation MIGHT still be physically wrong (e.g. K = mv² misses ½), but a dimensionally wrong equation is ALWAYS wrong!

💡 Simple Kid-Check: Dimensions act like a metal detector at the airport — if dimensions don't match, your formula is 100% illegal! If dimensions DO match, only the numeric factor (like ½ or 2π) might still need checking by experiment.

🔬 Deriving Simple Pendulum Time Period T:
Let T = k · mᵃ lᵇ gᶜ. Equating dimensions:
[M⁰ L⁰ T¹] = [Mᵃ] [Lᵇ] [LT⁻²]ᶜ = [Mᵃ Lᵇ⁺ᶜ T⁻²ᶜ]
• Comparing M: a = 0 (heavy or light bob takes the same time!)
• Comparing T: −2c = 1 ⇒ c = −½
• Comparing L: b + c = 0 ⇒ b = +½
Therefore T = k √(l / g) (experiment gives k = 2π).

Two classic 2-mark board questions: (1) Can a dimensionally correct equation be physically wrong? YES (T = √(l/g) misses 2π). (2) Why can't dimensional analysis derive s = ut + ½at²? Because it cannot handle equations with multiple added terms or trigonometric/exponential functions!

Solved Previous Year Questions (Kerala DHSE & NCERT)

Q1 (Kerala HSE Say • 1 Marks): Name two pairs of physical quantities having the same dimensional formula.

Key (1 Mark): (i) Work & Torque — both [M L² T⁻²]. (ii) Pressure & Stress (or Young's Modulus) — both [M L⁻¹ T⁻²]. (iii) Momentum & Impulse — both [M L T⁻¹].

Q2 (Kerala HSE March • 2 Marks): Check the dimensional correctness of the equation ½mv² = mgh.

LHS: [½mv²] = [M][LT⁻¹]² = [ML²T⁻²].
RHS: [mgh] = [M][LT⁻²][L] = [ML²T⁻²].
Since [LHS] = [RHS], the equation is dimensionally correct.

Q3 (NCERT / DHSE Model • 2 Marks): Each side of a cube is measured to be 7.203 m. Find the total surface area and volume of the cube to appropriate significant figures.

7.203 m has 4 sig figs.
• Surface Area = 6 × (7.203)² = 311.299... m² → 311.3 m² (4 sig figs; note 6 is exact).
• Volume = (7.203)³ = 373.714... m³ → 373.7 m³ (4 sig figs).

Q4 (Kerala HSE March • 3 Marks): The centripetal force F acting on a particle moving uniformly in a circle depends upon its mass m, velocity v, and radius r. Derive an expression for F using dimensional analysis.

Let F = k mᵃ vᵇ rᶜ.
[MLT⁻²] = [Mᵃ] [LT⁻¹]ᵇ [Lᶜ] = [Mᵃ Lᵇ⁺ᶜ T⁻ᵇ].
Equating exponents: a = 1, −b = −2 ⇒ b = 2, b + c = 1 ⇒ c = −1.
Hence F = k (mv² / r) (where experiment gives k = 1).

Q5 (Kerala HSE Say • 2 Marks): In the relation y = A sin(ωt − kx), where y & x are distances and t is time, find the dimensional formulae of ω and k.

Angles inside sin(...) are dimensionless [M⁰L⁰T⁰]:
• [ωt] = 1 ⇒ [ω] = [T⁻¹]
• [kx] = 1 ⇒ [k] = [L⁻¹]

Q6 (Kerala HSE March • 2 Marks): State any two limitations of dimensional analysis.

1. It cannot determine dimensionless numerical constants like ½ or 2π.
2. It cannot derive equations involving multiple added/subtracted terms (like s = ut + ½at²) or trigonometric/exponential/logarithmic functions.

Q7 (NCERT Exemplar / HSE • 2 Marks): A calorie is about 4.2 J where 1 J = 1 kg m² s⁻². Suppose we adopt a new unit system with mass = α kg, length = β m, time = γ s. Express 1 calorie in the new units.

n₂ = n₁ (M₁/M₂)¹ (L₁/L₂)² (T₁/T₂)⁻²
= 4.2 × (1 kg / α kg)¹ (1 m / β m)² (1 s / γ s)⁻² = 4.2 α⁻¹ β⁻² γ² new units.

Q8 (Kerala HSE Model • 1 Marks): "A dimensionally correct equation need not be physically exact, whereas a dimensionally wrong equation must be wrong." Justify with one example.

Correct! K = mv² is dimensionally right ([ML²T⁻²]) yet physically wrong by factor ½. Conversely, F = mv/t² has mismatched dimensions ([MLT⁻³] ≠ [MLT⁻²]) so it can NEVER represent force.