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Unit And Measurement Notes For Jee & Boards

Units and Measurements Class 11 Notes for JEE

PHYSICAL QUANTITIES

unit and dimension

In the context of physics, a physical quantity is a measurable property of a physical system. It is a characteristic that can be quantified or expressed numerically. Examples of physical quantities include length, mass, time, temperature, velocity, acceleration, and electric current, among others. Class 11 students typically study these fundamental physical quantities and their units, as well as the principles and laws that govern their behavior.

CLASSIFICATION OF PHYSICAL QUANTITY

Physical quantities can be classified into two main categories;
  • Fundamental quantity
  • Derived quantity

FUNDAMENTAL QUANTITY

These are basic physical quantities that cannot be define in the term of other quantities. They are independentand form the building blocks of other physical quantities. In the SI unit.
There are seven fundamental quantities :
Physical Quantity Unit Symbol
Length meter m
Mass kilogram kg
Time second s
Electric Current ampere A
Temperature kelvin K
Amount of Substance mole mol
Luminous Intensity candela cd

 AND SUPPLEMENTART UINT 

  • Radian (rad)
  • Steradian(Sr)

DERIVED QUANTITIES

These are quantities derived from one or more fundamental quantities through mathematical operations or combinations. They express a relationship between fundamental quantities. Some common derived quantities include :
  • Speed (meter per second, m/s): Derived from the fundamental quantities of length and time.
  • Acceleration (meter per second squared, m/s²): Derived from the fundamental quantities of length and time.
  • Force (newton, N): Derived from the fundamental quantities of mass, length, and time.
  • Energy (joule, J): Derived from the fundamental quantities of mass, length, and time.
  • Pressure (pascal, Pa): Derived from the fundamental quantities of force and area.
  • Electric Charge (coulomb, C): Derived from the fundamental quantities of electric current and time.
  • Power (watt, W): Derived from the fundamental quantities of energy and time.
  • ETC

UNITS

Units are standardized measures used to quantify and express physical quantities. They provide a consistent and universal way to describe and compare different quantities. In simple words, units are like labels that tell us how to measure and describe something.

For example, 
  • when we measure length, we use units such as meters (m) or centimeters (cm). 
  • When we measure time, we use units like seconds (s) or minutes (min). 
  • Units help us understand the magnitude or size of a quantity and allow us to make meaningful comparisons and calculations.

PROPERTIES OF UNITS

 Sure! Here are the properties of units summarized pointwise:

  •  Standardization: Units are standardized for consistency and universal acceptance.
  •  Invariability: Units remain the same regardless of location, time, or circumstances.
  •  Comparability: Units allow for the comparison of physical quantities.
  •  Scalability: Units can be multiplied or divided to represent larger or smaller quantities.
  •  Consistency: Units are consistent in usage across scientific disciplines.
  •  Interconvertibility: Units can be converted within a system of units using conversion factors.
  •  Dimensional Coherence: Units are consistent with the dimensions of the physical quantities they represent.
  •  Compatibility: Units can be added, subtracted, multiplied, or divided according to mathematical rules.
  •  Prefixes: Units can be modified with prefixes to represent multiples or submultiples.
  •  Accessibility: Units are widely accessible and used in various fields and everyday life.

Understanding these properties helps ensure accurate measurements and effective communication of scientific data.

 

SYSTEM OF UNITS

  •  International System of Units (SI): The SI system is the most widely used system of units and is internationally accepted. It is based on the metric system and provides standard units for measuring various physical quantities. The SI system consists of seven base units: meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, Kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity. Derived units are formed by combining these base units.
  •  Metric System: The metric system is a decimal-based system of units commonly used worldwide. It is closely related to the SI system and shares many base units with it. The metric system includes units such as meter (m) for length, gram (g) for mass, second (s) for time, and liter (L) for volume. The metric system uses prefixes such as kilo, centi, and milli to denote multiples or submultiples of the base unit.
  •  CGS System (Centimeter-Gram-Second): The CGS system is an older system of units primarily used in scientific research. It is based on the centimeter (cm) for length, gram (g) for mass, and second (s) for time as the fundamental units. Derived units in the CGS system include erg for energy, dyne for force, and poise for dynamic viscosity.
  •  FPS System (Foot-Pound-Second): The FPS system, also known as the British engineering system, was commonly used in the United States and the United Kingdom. It is based on the foot (ft) for length, pound (lb) for force or weight, and second (s) for time. The FPS system includes units such as foot (ft) for length, pound (lb) for force or weight, and second (s) for time. However, the FPS system is not widely used today, and the SI system has largely replaced it.
It's important to note that while the SI system is the most prevalent and widely accepted system of units, different systems may still be used in specific contexts or historical references.

DEFINITATION OF BASIC AND SUPPLEMENTARY UINTS

Fundamental Units (or base units) are the basic units of measurement used to quantify physical quantities. They are considered as the building blocks for constructing other units and cannot be derived from any other units. Fundamental units are chosen in such a way that they represent independent and fundamental properties of nature.

In the International System of Units (SI), there are seven fundamental units:

1. Meter (m): The meter is the fundamental unit of length. It is defined as the distance traveled by light in a vacuum during a specific fraction of a second.

2. Kilogram (kg): The kilogram is the fundamental unit of mass. It is defined as the mass of the international prototype of the kilogram, which is a platinum-iridium cylinder kept at the International Bureau of Weights and Measures in France.

3. Second (s): The second is the fundamental unit of time. It is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom.

4. Ampere (A): The ampere is the fundamental unit of electric current. It is defined as the constant current that, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one meter apart in a vacuum, would produce a force of exactly 2 × 10^−7 newtons per meter of length between the conductors.

5. Kelvin (K): The kelvin is the fundamental unit of temperature. It is defined as 1/273.16 of the thermodynamic temperature of the triple point of water. 

6. Mole (mol): The mole is the fundamental unit of amount of substance. It is defined as the amount of a substance that contains exactly 6.02214076 × 10^23 elementary entities (such as atoms, ions, or molecules).

7. Candela (cd): The candela is the fundamental unit of luminous intensity. It is defined as the luminous intensity in a given direction of a source that emits monochromatic radiation of frequency 540 × 10^12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

These fundamental units serve as the foundation for constructing derived units, which are combinations of fundamental units used to measure other physical quantities.


Supplementary Units:

Supplementary units, also known as supplementary dimensions or derived units of the second kind, are units used to measure physical quantities that are derived from the combination of base units. These units are defined in terms of the base units but are not considered fundamental units.

In the International System of Units (SI), there are two supplementary units:

1. Radian (rad): The radian is the supplementary unit for measuring plane angle. It is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. The radian is dimensionless, meaning it does not have any physical units associated with it.

2. Steradian (sr): The steradian is the supplementary unit for measuring solid angle. It is defined as the solid angle subtended at the center of a sphere by a surface area on the sphere that is equal to the square of the radius of the sphere. Like the radian, the steradian is also dimensionless.

Supplementary units, along with the base units and derived units, provide a comprehensive system for measuring various physical quantities. They are used in scientific calculations, engineering, and other fields where angles and solid angles are important measurements.

It's worth noting that while the radian and steradian are the primary supplementary units in the SI system, other unit systems may define additional supplementary units depending on the specific needs of that system.

DIMENSIONAL FORMULA

  • Dimensional formula is a mathematical expression that represents the dimensions of a physical quantity in terms of fundamental dimensions. It provides information about the nature and characteristics of a physical quantity without any numerical value. 
  • The dimensional formula of a quantity is obtained by expressing it as a product or quotient of the fundamental dimensions raised to certain powers. The fundamental dimensions in the International System of Units (SI) are length (L), mass (M), time (T), electric current (I), temperature (θ), amount of substance (N), and luminous intensity (J). Each of these dimensions is represented by a square bracket, such as [L] for length.
  • To obtain the dimensional formula, we assign the appropriate powers of the fundamental dimensions to the quantity's derived units or coefficients in a given equation or expression. By comparing the dimensions on both sides of the equation, we can verify its dimensional consistency.
  • The dimensional formula helps in understanding the relationships between physical quantities and deriving new formulas. It is also used for dimensional analysis, which involves examining the dimensions of different terms in an equation to check its correctness and derive relationships between physical quantities.
  • Overall, the dimensional formula serves as a useful tool in physics and other sciences, providing a systematic and standardized way to analyze and describe physical quantities in terms of their fundamental dimensions.

DIMENSIONAL ANALYSIS

  1. Dimensions: Dimensions represent the nature of a physical quantity and are expressed using square brackets, such as [L] for length, [M] for mass, and [T] for time.
  2.  Consistency: Dimensional analysis checks the dimensional consistency of equations to ensure that the dimensions of the terms on both sides of an equation match.
  3. Relationships: Dimensional analysis helps derive relationships between physical quantities based on their dimensions.
  4. Homogeneity: Dimensional analysis ensures homogeneity in equations by verifying that the units of measurement on both sides of an equation are compatible.
  5. Simplification: Dimensional analysis simplifies complex equations by reducing the number of variables and identifying key relationships.
  6. Unit Conversions: Dimensional analysis aids in converting units by utilizing conversion factors derived from the relationships between different dimensions.
  7. Predictions: Dimensional analysis can be used to make predictions about the behavior of physical quantities based on their dimensions and known relationships.
  8. Experimental Data Analysis: Dimensional analysis assists in analyzing experimental data by identifying the relevant dimensions and verifying the consistency of the obtained results.
  9. Problem-Solving: Dimensional analysis can be applied to problem-solving in physics, helping to determine the appropriate equation and unknown quantities based on their dimensions.
  10. Engineering Applications: Dimensional analysis is widely used in engineering disciplines to design and optimize systems, estimate parameters, and ensure the consistency of physical models.
In summary, dimensional analysis is a powerful tool in physics that examines the dimensions of physical quantities, checks the consistency of equations, derives relationships, simplifies calculations, aids in unit conversions, and has various applications in problem-solving and engineering.

CHARACTERISTICS OF DIMENSIONS

The characteristics of dimensions refer to the properties or attributes associated with dimensional quantities in physics. Here are the characteristics of dimensions:
  • Dimensions are independent: Dimensions of different physical quantities are independent of each other. Each physical quantity has its own set of dimensions that describe its nature and behavior.
  • Dimensions are algebraic quantities: Dimensions are treated as algebraic quantities that can be manipulated using mathematical operations such as addition, subtraction, multiplication, and division. They follow the same mathematical rules as variables.
  • Dimensions are represented by square brackets: In dimensional analysis, dimensions are denoted using square brackets ([ ]). For example, the dimension of length is represented as [L], the dimension of time as [T], and so on.
  • Dimensions can be added or subtracted: Dimensions of similar physical quantities can be added or subtracted. For example, the dimensions of length ([L]) can be added to the dimensions of length, resulting in [L] + [L] = [L]. Similarly, dimensions can be subtracted as well.
  • Dimensions can be multiplied or divided: Dimensions can be multiplied or divided according to the rules of algebra. For example, the dimensions of length ([L]) can be multiplied with the dimensions of time ([T]), resulting in [L] × [T] = [LT]. Similarly, dimensions can be divided as well.
  • Dimensions have exponentiation properties: Dimensions can be raised to a power or exponent. For example, the dimension of length ([L]) raised to the power of 2 is written as [L]^2.
  • Dimensions help in dimensional analysis: Dimensions play a crucial role in dimensional analysis, which is a method used to derive relationships between physical quantities and check the dimensional consistency of equations. By comparing the dimensions of different terms in an equation, we can identify if the equation is physically meaningful.

Understanding the characteristics of dimensions is important in physics as it allows for the analysis and manipulation of physical quantities using their dimensional properties.

PRINCIPAL OF HOMOGENEITY

According to the principle of homogeneity , The dimension of all ther terms that are given in a physical euation is must be equal to one another . 

For example :
Ex. s = ut + (½) at2

Dimensionally
[L] = [LT-1.T] + [LT-2. T2] [L] = [L] + [L]

 

DEFECT OF DIMENSIONAL ANALYSIS

Defects of Dimensional Analysis refer to the limitations or shortcomings of using dimensional analysis as a method in physics. At the NCERT level of Class 11, the following defects of dimensional analysis are commonly discussed:
  • Ignoring Dimensionless Constants: Dimensional analysis cannot determine dimensionless constants or coefficients that may appear in equations. These constants are crucial for obtaining accurate numerical values and cannot be determined through dimensional analysis alone.
  • Limited Information: Dimensional analysis provides information about the relationship between physical quantities based on their dimensions, but it does not provide detailed information about the underlying physics or specific numerical values. It does not account for the actual numerical factors that may affect the results.
  • Incomplete Representation: While dimensional analysis can be helpful in obtaining relationships between physical quantities, it may not capture all the factors that influence the phenomenon being studied. It may oversimplify complex physical systems and overlook important aspects that could affect the results.
  • Dependency on Units: Dimensional analysis relies on the choice of units for measurements. Different unit systems or conventions may yield different results in dimensional analysis. It is essential to use consistent units throughout the analysis to ensure accurate and meaningful results.
  • Neglecting Dimensionless Ratios: Dimensional analysis does not consider dimensionless ratios or quantities. These ratios often play a crucial role in determining physical behavior, and their exclusion can lead to incomplete or incorrect conclusions.
Despite these limitations, dimensional analysis remains a valuable tool in physics for establishing relationships, checking the dimensional consistency of equations, and gaining insights into the behavior of physical quantities. It is important to recognize the defects of dimensional analysis and use it judiciously in conjunction with other analytical methods to ensure accurate and comprehensive results.

FEW IMPORTANT DERIVED QUANTITIES


Derived Quantity Symbol Unit Dimensional Formula
Velocity v m/s [L][T]⁻¹
Acceleration a m/s² [L][T]⁻²
Force F N (Newton) [M][L][T]⁻²
Energy E J (Joule) [M][L]²[T]⁻²
Power P W (Watt) [M][L]²[T]⁻³
Pressure P Pa (Pascal) [M][L]⁻¹[T]⁻²
Momentum p kg·m/s [M][L][T]⁻¹
Angular Velocity ω rad/s [T]⁻¹
Angular Acceleration α rad/s² [T]⁻²
Impulse J N·s [M][L][T]⁻¹
Work W J (Joule) [M][L]²[T]⁻²
Frequency f Hz (Hertz) [T]⁻¹
Electric Charge Q C (Coulomb) [T][I]
Electric Potential V V (Volt) [M][L]²[T]⁻³[I]⁻¹
Resistance R Ω (Ohm) [M][L]²[T]⁻³[I]⁻²
Capacitance C F (Farad) [M]⁻¹[L]⁻²[T]⁴[I]²
Inductance L H (Henry) [M][L]²[T]⁻²[I]⁻²
Electric Current I A (Ampere) [I]
Magnetic Field Strength H A/m [I][L]⁻¹
Magnetic Flux Φ Wb (Weber) [M][L]²[T]⁻²[I]⁻¹
Resistivity ρ Ω·m [M][L]³[T]⁻³[I]⁻²
Electric Field Strength E N/C [M][L][T]⁻²[I]⁻¹
Heat Capacity C J/K [M][L]²[T]⁻²[θ]⁻¹
Stress σ N/m² [M][L]⁻¹[T]⁻²
Strain ε dimensionless NA
Young's Modulus Y N/m² [M][L]⁻¹[T]⁻²
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