Units and Measurements Class 11 Physics Notes

• Seven base units: metre (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount), candela (luminous intensity).
• Derived units are combinations of base units, e.g. newton (kg·m/s²), pascal (N/m²), joule (N·m).
• Common prefixes: milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), kilo (10³), mega (10⁶), giga (10⁹).

• Dimensional homogeneity: every term in a correct equation must have the same dimensions.
• Deriving relations: balance the dimensions of both sides and solve for powers.
• Converting units: use the dimensions as a conversion factor between unit systems.

• Systematic errors arise from instrument or method defects and are repeatable.
• Random errors arise from unpredictable fluctuations; they reduce with many measurements.
• Relative error:
• $\frac{\Delta a}{a_{\text{mean}}}$
• . Percentage error:
• $\frac{\Delta a}{a_{\text{mean}}}\times 100$
• .

• Addition/Subtraction: add the absolute errors.
• Multiplication/Division: add the relative errors.
• Raised to a power n: multiply the relative error by n.
• Combined:
• $\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$
• for
• $Z=\frac{A}{B}$
• .

• All non-zero digits are significant; zeros between non-zero digits are significant.
• Leading zeros are not significant; trailing zeros after a decimal point are.
• Multiplication/division: result keeps the fewest significant figures of any factor.
• Addition/subtraction: result keeps the fewest decimal places.
• Rules for rounding: round to the nearest, and for a 5 with preceding odd digit round up.

Example: Two readings of a length are $5.2\text{cm}$ and $5.4\text{cm}$. Find the absolute, relative and percentage error.

Solution: Mean $=(5.2+5.4)/2=5.3\text{cm}$. Absolute errors are 0.1 and 0.1. Mean absolute error $=0.1\text{cm}$. Relative error $=\frac{0.1}{5.3}\approx 0.019$; percentage error $\approx 1.9\%$.

Exercises (NCERT Q1.1–Q1.17)

• 1.1) Fill in the blanks: (a) The volume of a cube of side 1 cm is equal to … m³. (b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to … (mm)². (c) A vehicle moving with a speed of 18 km h⁻¹ covers … m in 1 s. (d) The relative density of lead is 11.3. Its density is … g cm⁻³ or … kg m⁻³.
• 1.2) Fill in the blanks by suitable conversion of units: (a) 1 kg m² s⁻² = … g cm² s⁻². (b) 1 m = … ly. (c) 3.0 m s⁻² = … km h⁻². (d) G = 6.67 × 10⁻¹¹ N m² (kg)⁻² = … (cm)³ s⁻² g⁻¹.
• 1.3) A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1 J = 1 kg m² s⁻². Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α⁻¹ β⁻² γ² in terms of the new units.
• 1.4) Explain this statement clearly: ‘To call a dimensional quantity large or small is meaningless without specifying a standard for comparison’. In view of this, reframe the following statements wherever necessary: (a) atoms are very small objects (b) a jet plane moves with great speed (c) the mass of Jupiter is very large (d) the air inside this room contains a large number of molecules (e) a proton is much more massive than an electron (f) the speed of sound is much smaller than the speed of light.
• 1.5) A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes 8 min and 20 s to cover this distance?
• 1.6) Which of the following is the most precise device for measuring length: (a) a vernier callipers with 20 divisions on the sliding scale (b) a screw gauge of pitch 1 mm and 100 divisions on the circular scale (c) an optical instrument that can measure length to within a wavelength of light?
• 1.7) A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair?
• 1.8) Answer the following: (a) You are given a thread and a metre scale. How will you estimate the diameter of the thread? (b) A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale? (c) The mean diameter of a thin brass rod is to be measured by vernier callipers. Why is a set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only?
• 1.9) The photograph of a house occupies an area of 1.75 cm² on a 35 mm slide. The slide is projected on to a screen, and the area of the house on the screen is 1.55 m². What is the linear magnification of the projector-screen arrangement?
• 1.10) State the number of significant figures in the following: (a) 0.007 m² (b) 2.64 × 10²⁴ kg (c) 0.2370 g cm⁻³ (d) 6.320 J (e) 6.032 N m⁻² (f) 0.0006032 m².
• 1.11) The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m and 2.01 cm respectively. Give the area and volume of the sheet to correct significant figures.
• 1.12) The mass of a box measured by a grocer’s balance is 2.30 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of the box, (b) the difference in the masses of the pieces, to the appropriate significant figures?
• 1.13) A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ m₀ of a particle in terms of its speed v and the speed of light, c. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes: m₀ = m(1 − v²)^(1/2). Guess where to put the missing c.
• 1.14) The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å: 1 Å = 10⁻¹⁰ m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m³ of a mole of hydrogen atoms?
• 1.15) One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large?
• 1.16) Explain this common observation clearly: If you look out of the window of a fast moving train, the nearby trees, houses etc. seem to move rapidly in a direction opposite to the train’s motion, but the distant objects (hill tops, the Moon, the stars etc.) seem to be stationary. (In fact, since you are aware that you are moving, these distant objects seem to move with you).
• 1.17) The Sun is a hot plasma (ionized matter) with its inner core at a temperature exceeding 10⁷ K, and its outer surface at a temperature of about 6000 K. At these high temperatures, no substance remains in a solid or liquid phase. In what range do you expect the mass density of the Sun to be, in the range of densities of solids and liquids or gases? Check if your guess is correct from the following data: mass of the Sun = 2.0 × 10³⁰ kg, radius of the Sun = 7.0 × 10⁸ m.

Answers at a glance

1. 1
   Q1.1 → (a) 10⁻⁶ m³ (b) 1.5 × 10⁴ mm² (c) 5 m (d) 11.3 g cm⁻³, 11.3 × 10³ kg m⁻³.
2. 2
   Q1.2 → (a) 10⁷ g cm² s⁻² (b) 1.06 × 10⁻¹⁶ ly (c) 3.9 × 10⁴ km h⁻² (d) 6.67 × 10⁻⁸ (cm)³ s⁻² g⁻¹.
3. 3
   Q1.3 → 1 calorie = 4.2 α⁻¹ β⁻² γ² in the new units — as required.
4. 4
   Q1.4 → Size, speed and mass are relative — every ‘large/small’ claim must name its comparison standard; (e) and (f) are already relative, others are reframed above.
5. 5
   Q1.5 → 500 new units of length.
6. 6
   Q1.6 → (c) The optical instrument is the most precise.
7. 7
   Q1.7 → Thickness of the hair ≈ 0.035 mm = 3.5 × 10⁻⁵ m.
8. 8
   Q1.8 → (a) Wind N turns and divide total length by N. (b) No — mechanical limits dominate. (c) 100 readings reduce random error of the mean far more than 5 readings do.
9. 9
   Q1.9 → Linear magnification ≈ 94.
10. 10
    Q1.10 → (a) 1 (b) 3 (c) 4 (d) 4 (e) 4 (f) 4.
11. 11
    Q1.11 → Area = 4.255 m²; volume = 8.55 × 10⁻² m³ (correct to significant figures).
12. 12
    Q1.12 → (a) 2.3 kg (b) 0.02 g.
13. 13
    Q1.13 → The missing c goes inside the bracket to make the argument dimensionless: m₀ = m(1 − v²/c²)¹/².
14. 14
    Q1.14 → Total atomic volume of a mole of hydrogen ≈ 3.15 × 10⁻⁷ m³.
15. 15
    Q1.15 → Ratio ≈ 7 × 10⁴; large because a gas is mostly empty space — molecules are far apart compared with their sizes.
16. 16
    Q1.16 → Nearby objects sweep large angles quickly; distant objects sweep negligible angles, so they appear stationary.
17. 17
    Q1.17 → ρ ≈ 1.4 × 10³ kg m⁻³ — in the range of solids and liquids, confirming the guess.

MCQ Questions

1 Mark Questions

1. 1
   Name the SI base units of time and luminous intensity.
2. 2
   State the principle of homogeneity of dimensions.
3. 3
   Define the least count of a measuring instrument.
4. 4
   What is the dimensional formula of work?

2 Mark Questions

1. 1
   Check the dimensional consistency of the equation v = u + at. (Ans: [L T⁻¹] = [L T⁻¹] + [L T⁻¹], dimensionally correct)
2. 2
   Check whether the equation s = ut + ½at² is dimensionally correct (u: velocity, a: acceleration, s: distance, t: time). (Ans: both sides have the dimension of length [L], hence correct)
3. 3
   A quantity X is given by X = a²b³/c. If the percentage errors in a, b and c are 1%, 2% and 3%, find the percentage error in X. (Ans: 2×1 + 3×2 + 3 = 11%)

3 Mark Questions

1. 1
   Using dimensional analysis, derive the relation for the period of a simple pendulum (T ∝ √(l/g)). (Ans: T = k(l/g)^(1/2))
2. 2
   State the rules governing the number of significant figures in a result, with one example of addition and one of multiplication.
3. 3
   Explain how a screw gauge works and write the formula for its least count. (Ans: LC = pitch/number of divisions on circular scale)

5 Mark Questions

1. 1
   Describe the construction and use of a vernier callipers. Explain how the least count is found and how a measurement with zero error is corrected, with one worked reading.
2. 2
   Explain the parallax method for measuring the distance of a far-off star. Also derive the use of dimensions to check the correctness of s = ut + ½at² and v² = u² + 2as.