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Physics · Class 11 Science

Ch 1Units and Measurement — Class 11 Physics, concept-first.

Physics rests on measurement. Every law, every experiment, and every observation in physics ultimately comes down to measuring something — a length, a time interval, a mass. But what does "measuring a physical quantity" actually mean?

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Key concepts

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Significant Figures Calculation

Imagine you're measuring the length of a table with a ruler that has marks every millimeter. You see the table edge falls somewhere between 152.3 cm and 152.4 cm. You estimate it as 152.35 cm.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1.1

Introduction

Physics rests on measurement. Every law, every experiment, and every observation in physics ultimately comes down to measuring something — a length, a time interval, a mass.

+Worked Examplesi1 question
  1. Example 1.1Each side of a cube is measured to be $7.203\ \text{m}$. What are the total surface area and the volume of the cube to appropriate significa…Preview
1.2

The International System of Units

Before the modern era, scientists in different countries used different systems of units. This created confusion when comparing experimental results across borders.

+Worked Examplesi1 question
  1. Example 1.2$5.74\ \text{g}$ of a substance occupies $1.2\ \text{cm}^3$. Express its density by keeping the significant figures in view.Preview
1.3

Significant Figures

Every measurement carries some error. When you report a measured value, you must communicate how precise that measurement actually is.

+Worked Examplesi1 question
  1. Example 1.3Let us consider an equation $\dfrac{1}{2}\,m v^2 = m g h$ where $m$ is the mass of the body, $v$ its velocity, $g$ is the acceleration due t…Preview
1.3.1

Rules for Arithmetic Operations with Significant Figures

When you perform calculations with measured values, the result cannot be more precise than the least precise measurement you started with.

1.3.2

Rounding Off Uncertain Digits

When a computation produces a number with more digits than justified, you must round it off. The rules are:

1.3.3

Rules for Determining Uncertainty in Arithmetic Results

When you multiply measured values, the percentage uncertainties add.

1.4

Dimensions of Physical Quantities

Every physical quantity has a nature — it is not just a number, but a kind of thing. Length, mass, and time are fundamentally different from each other. This "kind" is what we call its dimension.

+Worked Examplesi1 question
  1. Example 1.4The SI unit of energy is $J = \text{kg m}^2\,\text{s}^{-2}$; that of speed $v$ is $\text{m s}^{-1}$ and of acceleration $a$ is $\text{m s}^{…Preview
1.5

Dimensional Formulae and Dimensional Equations

Every physical quantity can be expressed in terms of the seven base quantities (mass, length, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity).

+Worked Examplesi1 question
  1. Example 1.5Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the pe…Preview
1.6

Dimensional Analysis and its Applications

The most fundamental idea in dimensional analysis is that you can only add or subtract physical quantities that have the same dimensions. This is not a convention — it is a logical necessity.

1.6.1

Checking the Dimensional Consistency of Equations

When you write an equation, every term that is added or subtracted must have the same dimensions. Consider the equation for the distance travelled by a particle starting from position with initial vel…

1.6.2

Deducing Relations Among Physical Quantities

The method of dimensions can sometimes be used to derive the form of a relationship between physical quantities, provided you know which quantities the dependent variable depends on.

Summary

- Physical quantities are expressed as a product of a numerical value and a unit: . - Fundamental quantities (mass, length, time, etc.) have base SI units (kg, m, s, A, K, mol, cd).

Exercises

+Show 17 questions17 questions
  1. 1.1Fill in the blanks: (a) The volume of a cube of side 1 cm is equal to ..... $\text{m}^3$. (b) The surface area of a solid cylinder of radius…Free
  2. 1.2Fill in the blanks by suitable conversion of units: (a) $1\ \text{kg m}^2\,\text{s}^{-2} = \ldots\ \text{g cm}^2\,\text{s}^{-2}$ (b) $1\ \te…Free
  3. 1.3A calorie is a unit of heat (energy in transit) and it equals about $4.2\ \text{J}$ where $1\,\text{J} = 1\ \text{kg m}^2\,\text{s}^{-2}$. S…Free
  4. 1.4Explain this statement clearly: "To call a dimensional quantity 'large' or 'small' is meaningless without specifying a standard for comparis…Preview
  5. 1.5A 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…Preview
  6. 1.6Which of the following is the most precise device for measuring length: (a) a vernier callipers with 20 divisions on the sliding scale (b) a…Preview
  7. 1.7A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and fi…Preview
  8. 1.8Answer 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…Preview
  9. 1.9The photograph of a house occupies an area of $1.75\ \text{cm}^2$ on a 35 mm slide. The slide is projected on to a screen, and the area of t…Preview
  10. 1.10State the number of significant figures in the following: (a) $0.007\ \text{m}^2$ (b) $2.64 \times 10^{24}\ \text{kg}$ (c) $0.2370\ \text{g…Preview
  11. 1.11The length, breadth and thickness of a rectangular sheet of metal are $4.234\ \text{m}$, $1.005\ \text{m}$, and $2.01\ \text{cm}$ respective…Preview
  12. 1.12The mass of a box measured by a grocer's balance is $2.30\ \text{kg}$. Two gold pieces of masses $20.15\ \text{g}$ and $20.17\ \text{g}$ are…Preview
  13. 1.13A famous relation in physics relates 'moving mass' $m$ to the 'rest mass' $m_0$ of a particle in terms of its speed $v$ and the speed of lig…Preview
  14. 1.14The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å: $1\ \text{Å} = 10^{-10}\ \text{m}$. The size…Preview
  15. 1.15One mole of an ideal gas at standard temperature and pressure occupies $22.4\ \text{L}$ (molar volume). What is the ratio of molar volume to…Preview
  16. 1.16Explain this common observation clearly: If you look out of the window of a fast moving train, the nearby trees, houses etc. seem to move ra…Preview
  17. 1.17The Sun is a hot plasma (ionized matter) with its inner core at a temperature exceeding $10^7\ \text{K}$, and its outer surface at a tempera…Preview

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 44 questions44 questions
  1. Q1The number of significant figures in 0.06900 is (a) 5 (b) 4 (c) 2 (d) 3Free
  2. Q2The sum of the numbers 436.32, 227.2 and 0.301 in appropriate significant figures is (a) 663.821 (b) 664 (c) 663.8 (d) 663.82Free
  3. Q3The mass and volume of a body are 4.237 g and 2.5 cm$^3$, respectively. The density of the material of the body in correct significant figur…Free
  4. Q4The numbers 2.745 and 2.735 on rounding off to 3 significant figures will give (a) 2.75 and 2.74 (b) 2.74 and 2.73 (c) 2.75 and 2.73 (d) 2.7…Preview
  5. Q5The length and breadth of a rectangular sheet are 16.2 cm and 10.1 cm, respectively. The area of the sheet in appropriate significant figure…Preview
  6. Q6Which of the following pairs of physical quantities does not have same dimensional formula? (a) Work and torque. (b) Angular momentum and Pl…Preview
  7. Q7Measure of two quantities along with the precision of respective measuring instrument is $A = 2.5$ m s$^{-1} \pm 0.5$ m s$^{-1}$, $B = 0.10$…Preview
  8. Q8You measure two quantities as $A = 1.0$ m $\pm 0.2$ m, $B = 2.0$ m $\pm 0.2$ m. We should report correct value for $\sqrt{AB}$ as: (a) $1.4$…Preview
  9. Q9Which of the following measurements is most precise? (a) 5.00 mm (b) 5.00 cm (c) 5.00 m (d) 5.00 km.Preview
  10. Q10The mean length of an object is 5 cm. Which of the following measurements is most accurate? (a) 4.9 cm (b) 4.805 cm (c) 5.25 cm (d) 5.4 cmPreview
  11. Q11Young's modulus of steel is $1.9 \times 10^{11}$ N/m$^2$. When expressed in CGS units of dynes/cm$^2$, it will be equal to ($1$ N $= 10^5$ d…Preview
  12. Q12If momentum ($P$), area ($A$) and time ($T$) are taken to be fundamental quantities, then energy has the dimensional formula (a) $(P^1 A^{-1…Preview
  13. Q13(MCQ II — one or more options correct) On the basis of dimensions, decide which of the following relations for the displacement of a particl…Preview
  14. Q14(MCQ II — one or more options correct) If $P, Q, R$ are physical quantities, having different dimensions, which of the following combination…Preview
  15. Q15(MCQ II — one or more options correct) Photon is quantum of radiation with energy $E = h\nu$ where $\nu$ is frequency and $h$ is Planck's co…Preview
  16. Q16(MCQ II — one or more options correct) If Planck's constant ($h$) and speed of light in vacuum ($c$) are taken as two fundamental quantities…Preview
  17. Q17(MCQ II — one or more options correct) Which of the following ratios express pressure? (a) Force/Area (b) Energy/Volume (c) Energy/Area (d)…Preview
  18. Q18(MCQ II — one or more options correct) Which of the following are not a unit of time? (a) Second (b) Parsec (c) Year (d) Light yearPreview
  19. Q19Why do we have different units for the same physical quantity?Preview
  20. Q20The radius of atom is of the order of 1 Å and radius of nucleus is of the order of fermi. How many magnitudes higher is the volume of atom a…Preview
  21. Q21Name the device used for measuring the mass of atoms and molecules.Preview
  22. Q22Express unified atomic mass unit in kg.Preview
  23. Q23A function $f(\theta)$ is defined as: $f(\theta) = 1 - \theta + \dfrac{\theta^2}{2!} - \dfrac{\theta^3}{3!} + \dfrac{\theta^4}{4!}$. Why is…Preview
  24. Q24Why length, mass and time are chosen as base quantities in mechanics?Preview
  25. Q25(a) The earth-moon distance is about 60 earth radius. What will be the diameter of the earth (approximately in degrees) as seen from the moo…Preview
  26. Q26Which of the following time measuring devices is most precise? (a) A wall clock. (b) A stop watch. (c) A digital watch. (d) An atomic clock.…Preview
  27. Q27The distance of a galaxy is of the order of $10^{25}$ m. Calculate the order of magnitude of time taken by light to reach us from the galaxy…Preview
  28. Q28The vernier scale of a travelling microscope has 50 divisions which coincide with 49 main scale divisions. If each main scale division is 0.…Preview
  29. Q29During a total solar eclipse the moon almost entirely covers the sphere of the sun. Write the relation between the distances and sizes of th…Preview
  30. Q30If the unit of force is 100 N, unit of length is 10 m and unit of time is 100 s, what is the unit of mass in this system of units?Preview
  31. Q31Give an example of (a) a physical quantity which has a unit but no dimensions. (b) a physical quantity which has neither unit nor dimensions…Preview
  32. Q32Calculate the length of the arc of a circle of radius 31.0 cm which subtends an angle of $\dfrac{\pi}{6}$ at the centre.Preview
  33. Q33Calculate the solid angle subtended by the periphery of an area of 1 cm$^2$ at a point situated symmetrically at a distance of 5 cm from the…Preview
  34. Q34The displacement of a progressive wave is represented by $y = A \sin(\omega t - k x)$, where $x$ is distance and $t$ is time. Write the dime…Preview
  35. Q35Time for 20 oscillations of a pendulum is measured as $t_1 = 39.6$ s; $t_2 = 39.9$ s; $t_3 = 39.5$ s. What is the precision in the measureme…Preview
  36. Q36A new system of units is proposed in which unit of mass is $\alpha$ kg, unit of length $\beta$ m and unit of time $\gamma$ s. How much will…Preview
  37. Q37The volume of a liquid flowing out per second of a pipe of length $l$ and radius $r$ is written by a student as $v = \dfrac{\pi}{8}\dfrac{P…Preview
  38. Q38A physical quantity $X$ is related to four measurable quantities $a$, $b$, $c$ and $d$ as follows: $X = a^2 b^3 c^{5/2} d^{-2}$. The percent…Preview
  39. Q39In the expression $P = E l^2 m^{-5} G^{-2}$, $E$, $m$, $l$ and $G$ denote energy, mass, angular momentum and gravitational constant, respect…Preview
  40. Q40If velocity of light $c$, Planck's constant $h$ and gravitational constant $G$ are taken as fundamental quantities then express mass, length…Preview
  41. Q41An artificial satellite is revolving around a planet of mass $M$ and radius $R$, in a circular orbit of radius $r$. From Kepler's Third law…Preview
  42. Q42In an experiment to estimate the size of a molecule of oleic acid 1 mL of oleic acid is dissolved in 19 mL of alcohol. Then 1 mL of this sol…Preview
  43. Q43(a) How many astronomical units (A.U.) make 1 parsec? (b) Consider a sunlike star at a distance of 2 parsecs. When it is seen through a tele…Preview
  44. Q44Einstein's mass-energy relation emerging out of his famous theory of relativity relates mass ($m$) to energy ($E$) as $E = mc^2$, where $c$…Preview