NCERT Class 11 Physics Chapter 2 Notes Units and Measurement - Download PDF

NCERT Class 11 Physics Chapter 1 Notes: Download PDF

Chapter 1 Units and Measurements lays the foundation of physics by describing physical quantities, SI units, significant figures, errors and dimensional analysis. These Units and Measurement Class 11 Notes consist of brief explanations and solved problems, and so are ideal when revising physics in a short time and for last-minute preparations for exams.

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NCERT Class 11 Physics Chapter 1 Notes

NCERT Class 11 Physics Chapter 1 Notes Units and Measurement introduce the foundation of Physics by explaining how physical quantities are measured, the importance of SI units, and the role of significant figures. These notes simplify concepts like accuracy, precision, and dimensional analysis, making them very useful for quick revision and exam preparation.

Physical Quantity

• A physical quantity is something we can measure, and it helps us understand and describe different things that happen in the world through rules. Examples of physical quantities are length, mass, time, and force.

Physical quantity (Q) = Magnitude × Unit = n × u

Types of Physical Quantity

1. Ratio (Numerical Value Only): A physical quantity has no unit when it is a ratio of two similar quantities.
   Example: Relative density, refractive index, and strain are a few examples.
2. Scalar (Only Magnitude): Quantities such as length, time, work, and energy have no direction. Without specifying a direction, the magnitude can be negative, indicating a negative numerical value. Ordinary addition and subtraction laws apply to scalar quantities.
3. Vector (Magnitude and Direction): Vector quantities include displacement, velocity, acceleration, and force. Vector laws of addition, as opposed to ordinary addition laws, govern vectors.

Fundamental Quantities and Fundamental Units

• The units of quantities ( length, mass, and time) are independent of one another; no one can be changed or related to any other unit. These quantities are called fundamental quantities and their units are called fundamental units.
• Seven fundamental quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminescent intensity.

Derived Quantities and Derived Units:

• Derived Quantities are products and ratios of the fundamental quantities that exist in a system of units and these quantities can be expressed in terms of other basic quantities.
• Units of physical quantities derived from the seven fundamental quantities are termed "derived units." Examples include:

Volume = length ✕ breadth ✕ height = m ✕ m ✕ m = m³

Density = mass/volume = kgm⁻³

The International System of Units

A unit system is a comprehensive set of units that includes both fundamental and derived units for various physical quantities. Typical systems include:

• CGS system: The Gaussian system of units, also known as the CGS system, designates length, mass, and time as fundamental quantities, with corresponding units of centimetre (cm), gramme (g), and second (s).
• FPS system: Foot, pound, and second are the units of length, mass, and time in the FPS system, respectively. In this system, force is a derived quantity with the poundal as its unit.
• MKS system:- The Giorgi system, also known as the MKS system, designates length, mass, and time as fundamental quantities, with corresponding fundamental units of length (m), mass (kg), and time (s).
• S.I. system:- It is known as the International System of Units, which was accepted in 1960 and it is an extended system of units that is used throughout physics. This system has seven fundamental quantities. The quantities and units are listed in the table below.

Basic Units

Supplementary Units

Practical Units

Practical units of length:

1 kilometer(km) = 10³m

1 centimeter(cm) = 10⁻²m

1 millimeter (mm) = 10⁻³m

1 micron(μ) = 10⁻⁶m

1 nanometer (nm) = 10⁻⁹m

1 Angstrom (Å) = 10⁻¹ºm

1 fermometer = 1 fermi (fm) = 10⁻¹⁵m

Astronomical unit (AU): 1AU is the average distance of the sun from the Earth, commonly used for measuring astronomical distances.

1 AU = 1.496 ✕ 10¹¹m ≅ 1.5 ✕ 10¹¹m

Light Year (ly): 1 light-year is the distance travelled by light in one year.

1 light-year = (3 ✕ 10⁸ms⁻¹) ✕ (365 ✕24✕60✕60)s = 9.46 ✕ 10¹⁵m

Parallactic second or parsec(pc): The parallactic second (pc): 1 parsec is the distance between the average radius of the Earth's orbit around the Sun and an angle of 1" (second of arc).

$\begin{array}{r}1\text{parsec}=\frac{1AU}{1^{\prime \prime }}\\ 1AU=1.496\times 0h{10}^{11}\mathrm{m}\\ 1^{\prime \prime }=\left(\frac{1}{3600}\right)\times \frac{\pi }{{180}^{\circ }}=4.85\times {10}^{-6}\mathrm{rad}\\ \therefore 1\text{par sec}=\frac{1.496\times {10}^{11}\mathrm{m}}{4.85\times {10}^{-6}\mathrm{rad}}=3.084\times {10}^{16}\mathrm{m}\\ 1pc=3.084\times {10}^{16}\mathrm{m}\approx 3.1\times {10}^{16}\mathrm{m}\\ 1pc=3.261\mathrm{y}\end{array}$

Practical units of mass:

1 metric ton = 10³kg

1 quintal = 10²kg

1 gram = 10⁻³kg

1miligram = 10⁻³g = 10⁻⁶kg

To measure the mass of an atom or molecule, we use a unified atomic mass unit(u) defined as 1/12th the mass of an atom of the carbon-12 isotope.

1u = 1.66 ✕ 10⁻²⁷kg

Practical units of time:

1 millisecond (ms) = 10⁻³s

1 microsecond (*s) = 10⁻⁶s

1 nanosecond (ns) = 10⁻⁹s

1 picosecond (ps) = 10⁻¹²s

A shake is the unit of time used in microscopic physics

1 shake = 10⁻⁸s

Significant Figures

Significant figures in a physical quantity's measured value indicate the number of digits in which we have confidence. The greater the number of significant figures obtained in a measurement, the greater the measurement's accuracy. The opposite is also true.

When counting the number of significant figures in a given measured quantity, the following rules apply.

(1) Non-zero digits: A number's non-zero digits are all considered significant.

For example, 41.3 contains three significant figures, 147.6 contains four significant figures, and 12.123 contains five significant figures.

(2) Between non-zero digits, a zero: Significant is A zero between two non-zero digits.

For example, 5.03 contains three significant figures, 5.404 contains four significant figures, and 6.004 contains four significant figures.

(3) Leading zeros: Leading zeros (zeroes to the left of the first non-zero digit) are ignored.

For example, the number 0.583 has three significant figures, the number 0.045 has two significant figures, and the number 0.003 has one significant figure.

(4) Trailing zeros: Trailing zeros are considered significant (zeroes to the right of the last non-zero digit).

For example, 4.250 has four significant figures, 434.00 has five significant figures, and 243.000 has six significant figures.

(5) Exponential notation: The numerical portion of exponential notation indicates the number of significant figures.

For example, 1.32 10*10^-2 has three significant figures, while 1.32*10⁴ has three significant figures.

Rounding Off

When rounding off measurements, the following rules are followed:

• If the digit to be dropped is less than 5, the preceding digit is not affected.

For example, 7.82 is rounded to 7.8, and 3.94 is rounded to 3.9.

• If the digit to be dropped is greater than 5, the digit before it is increased by one.

For example, 6.87 is rounded to 6.9, and 12.78 is rounded to 12.8.

• If the digit to be dropped is 5 followed by a non-zero digit, the preceding digit is increased by one.

For example, 16.351 is rounded to 16.4, and 6.758 is rounded to 6.8.

• If the dropped digit is 5 or 5 followed by zeros and the preceding digit is even, it is dropped.

For example, 3.250 becomes 3.2, and 12.650 becomes 12.6.

• If the digit to be dropped is 5 or 5 followed by zeros and the preceding digit is odd, the digit to be dropped is increased by one.

For example, 3.750 is rounded to 3.8, and 16.150 is rounded to 16.2.

Dimensions of Physical Quantities

All derived physical quantities can be expressed using the seven base quantities as discussed at the beginning of this chapter. Hence, these quantities are also referred to as the seven dimensions of the physical world. They are represented by using square brackets. Thus, the seven dimensions of the physical world are represented as follows.

[M] for mass
[L] for length
[T] for time
[A] for electric current
[K] for thermodynamic temperature
[cd] for luminous intensity
[mol] for amount of substance

The dependence of all other physical quantities on these base quantities can be expressed in terms of their dimensions.
For example, the speed $v$, of a moving object is given by the relation $v=\frac{\text{Distance}}{\text{Time}}$
Therefore, the dimension of speed can be given as

$[v]=\frac{[\mathrm{L}]}{[\mathrm{T}]}=\left[{\mathrm{LT}}^{-1}\right]$

Thus, the dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.

Dimensional Formulae and Dimensional Equations

The expression of a physical quantity in terms of its dimensions is called its dimensional formula. For example, the dimensional formula of force is $\left[\mathrm{MLT}{\mathrm{T}}^{-2}\right]$ and that for acceleration is $\left[{\mathrm{M}}^0{\mathrm{LT}}^{-2}\right]$. Dimensional formula for density is $\left[{\mathrm{ML}}^{-3}{\mathrm{T}}^0\right]$.
The table given in the formulae chart at the end of this chapter gives the dimensional formulae for some physical quantities.
An equation which contains a physical quantity on one side and its dimensional formula on the other side is called the dimensional equation of that quantity. Dimensional equations for a few physical quantities are given below.
Speed $[V]=\left[M^0{L}^{-1}\right]$
Area $[A]=\left[M^0{L}^2{T}^0\right]$
Force $[F]=\left[{\mathrm{MLT}}^{-2}\right]$ etc.
The physical quantities having the same derived units have the same dimensions.

Dimensional Analysis and Its Applications

The dimensions of a physical quantity are calculated by raising the fundamental units to the powers that are needed to derive its derived units.

The length, mass, and time are denoted by [L], [M], and [T], respectively.

Uses of Dimensional Equations

• To convert units of one system into units of another system: The product of a physical quantity's numerical value and its corresponding unit is a constant. Let the numerical value of a physical quantity p be n₁ and n₂ in two different systems and the corresponding units be u₁ and u₂, then
  
  $p=n_1\left(u_1\right)=n_2\left(u_2\right)-(\mathrm{i})$

If the dimensions of the physical quantity are a in mass, b in length, and c in time, then its dimensional formula will be $\left[M_1^a{L}_1^b{T}_1^c\right]$. If the fundamental units in one system are M₁, L₁, and T₁

$p=n_1\left(M_1^a{L}_1^b{T}_1^c\right)$

Similarly, if the fundamental units in the second system are M₂, L₂, and T₂, then

$p=n_2\left(M_2^a{L}_2^b{T}_2^c\right)$

According to Eqn (i), we have

$\begin{array}{r}{n}_1\left(M_1^a{L}_1^b{T}_1^c\right)=n_2\left(M_2^a{L}_2^b{T}_2^c\right)\\ \therefore n_2=n_1{\left(\frac{M_1}{M_2}\right)}^a{\left(\frac{L_1}{L_2}\right)}^b{\left(\frac{T_1}{T_2}\right)}^c\end{array}$

Using this formula, we can convert the numerical value of a physical quantity from one system of units to another system.

• To check the correctness of an equation: All terms on both sides of a physical equation must have the same dimensions. This is known as the dimension homogeneity principle.

$\begin{array}{r}\text{E.g.,}\frac{1}{2}m{v}^2=mgh\frac{1}{2}m{v}^2=mgh\\ \mathrm{m}=[\mathrm{M}],\mathrm{v}=\left[{\mathrm{LT}}^{-1}\right],\mathrm{g}=\left[{\mathrm{LT}}^{-2}\right]\text{and}\mathrm{h}=[\mathrm{L}]\\ [\mathrm{M}]{\left[{\mathrm{LT}}^{-1}\right]}^2=[\mathrm{M}]\left[{\mathrm{LT}}^{-2}\right][\mathrm{L}]\\ \left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]=\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]\end{array}$

Limitations:

• Cannot give numerical constants (like $1/2,\pi$, etc.).
• Not applicable to equations involving logarithms, trigonometric or exponential functions.
• Works only for dimensionally homogeneous equations.

Units and Measurements: Previous Year Question and Answer

Q1:

The length and breadth of a rectangular sheet are 16.2 cm and 10.1 cm respectively. The area of the sheet in appropriate significant figures and error is:

(a) $164 \pm 3 \mathrm{~cm}^2$

(b) $163.62 \pm 2.6 \mathrm{~cm}^2$

(c) $163.6 \pm 2.6 \mathrm{~cm}^2$

(d) $163.62 \pm 3 \mathrm{~cm}^2$

Answer:

Given :

$$
\begin{aligned}
& l=16.2 \mathrm{~cm}, \text { and } \Delta l=0.1 \\
& b=10.1 \mathrm{~cm}, \Delta b=0.1 \\
& l=16.1 \pm 0.1 \text { and } b=10.1 \pm 0.1
\end{aligned}
$$

$$
\begin{aligned}
\text { Now, area (a) } & =l \times b \\
& =16.2 \times 10.1 \\
& =163.62 \mathrm{~cm}^2 \\
& =164 \mathrm{~cm}^2
\end{aligned}
$$

(by rounding off the answer of the area in 3 significant numbers)

$$
\begin{aligned}
& \Delta \frac{A}{A}=\frac{\Delta l}{l}+\frac{\Delta b}{b} \\
& =\frac{0.1}{16.2}+\frac{0.1}{10.1} \\
& \frac{\Delta A}{164}=\left(\frac{10.1 \times 0.1+16.2 \times 0.1}{16.2 \times 10.1}\right)
\end{aligned}
$$

Thus, $\Delta A=2.64 \mathrm{~cm}^2$
We get $\Delta A=3 \mathrm{~cm}^2$
(by rounding off the answer of $\Delta A$ in 1 significant number)
Thus, $\Delta A=(164 \pm 3) \mathrm{cm}^2$

Hence, the correct answer is option (a).

Q2:

Which of the following are not units of time?

a) second

b) parsec

c) year

d) light year

Answer:

Parsec and Light year
Explanation: Parsec & Light year are units to measure distance, while second \& Year are units to measure time.

Hence, the answers are options (b) and (d).

Q3:

If momentum (P), area (A), and time (T) are taken to be fundamental quantities, then energy has the dimensional formula

(a) $\left(P^1 A^{-1} T^1\right)$

(b) $\left(P^2 A^1 T^1\right)$

(c) $\left(P^1 A^{-1 / 2} T^1\right)$

(d) $\left(P^1 A^{1 / 2} T^{-1}\right)$

Answer:

Explanation: Let us consider $\left[P^a A^b T^c\right]$ the formula for energy for fundamental quantities $\mathrm{P}, \mathrm{A}$ \& T .
Thus, the dimensional formula of-

$
\begin{aligned}
& \text { Energy }(\mathrm{E})=\left[P^a A^b T^c\right] \\
& \text { Momentum }(\mathrm{P})=\left[M L T^{-1}\right] \\
& \text { Area }(\mathrm{A})=\left[L^2\right] \\
& \operatorname{Time}(\mathrm{T})=\left[T^1\right]
\end{aligned}
$

Now, E = f.s

Thus, $\left[P^1 A^{1 / 2} T^{-1}\right]$ is the dimensional formula of energy.

Hence, the answer is option (d).

Importance of Units and Measurement Class 11 Notes

Origin of Physics Concepts

• All physics calculations are based on Units and measurements. Without a standard unit system, it is impossible to compare such physical entities as length, mass, and time.

Precision with Numerical Problems

• The notes Class 11 Physics Units and Measurement are used to assist students in solving numerical questions with accurate precision, which is very important in CBSE board exams, JEE and NEET.

Learning SI Units and Conversions

• These notes describe the International System of Units (SI units), dimensional formula, and unit conversions, which are important to higher physics and engineering exams and problem-solving.

Significant Figures and Analysis of Error.

• This chapter also emphasises the importance of important figures, errors and approximations in real-life measurements, and so the students are more assured in their approach to experiments and questions in the lab.

Connection of Theory and Experiments

• Units and Measurement provides an intermediate between theoretical physics and the actual experiment since it instructs the correct approach to recording, calculating, and analysing the physical data.

Boosts Exam Preparation

• Properly organised NCERT Class 11 Physics Notes on Units and Measurement are handy both in last-minute revision and in enabling students to have a higher score in the exams.

Competitive Exam Relevance

• The chapter is commonly referred to in JEE Main, NEET and Olympiad tests. The ability to master these notes will enhance the speed and accuracy of a student in solving problems during competitive exams.

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