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

NCERT Notes for Class 11 Chapter 1: Download PDF

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NCERT Notes for Class 11 Chapter 1

The NCERT Notes of CBSE Class 11 Physics Chapter 1 - Units and Measurements provide a brief and neat overview of important principles, equations and procedures connected to the physical quantities and measurements. The notes are perfect to study and revise the topics on CBSE, JEE and NEET.

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).

$\begin{aligned} & 1 \text { parsec }=\frac{1 A U}{1^{\prime \prime}} \\ & 1 A U=1.496 \times 0 h 10^{11} \mathrm{~m} \\ & 1^{\prime \prime}=\left(\frac{1}{3600}\right) \times \frac{\pi}{180^{\circ}}=4.85 \times 10^{-6} \mathrm{rad} \\ & \therefore 1 \text { par sec }=\frac{1.496 \times 10^{11} \mathrm{~m}}{4.85 \times 10^{-6} \mathrm{rad}}=3.084 \times 10^{16} \mathrm{~m} \\ & 1 p c=3.084 \times 10^{16} \mathrm{~m} \approx 3.1 \times 10^{16} \mathrm{~m} \\ & 1 p c=3.261 \mathrm{y}\end{aligned}$

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 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

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 in the beginning of this chapter. Hence these quantities are also referred as seven dimensions of physical world. They are represented by using square brackets. Thus the seven dimensions of 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 $\quad[V]=\left[M^0 L^{-1}\right]$
Area $\quad[A]=\left[M^0 L^2 T^0\right]$
Force $\quad[F]=\left[\mathrm{MLT}^{-2}\right]$ etc.
The physical quantities having same derived units have 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 are n₁ and n₂ in two different systems and the corresponding units are 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

$\begin{aligned} & n_1\left(M_1^a L_1^b T_1^c\right)=n_2\left(M_2^a L_2^b T_2^c\right) \\ & \therefore n_2=n_1\left(\frac{M_1}{M_2}\right)^a\left(\frac{L_1}{L_2}\right)^b\left(\frac{T_1}{T_2}\right)^c\end{aligned}$

Using this formula we can convert the numerical value of a physical quantity from one system of units into the other 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.

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

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.

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