Units and Measurement: 2024-25

• The session is the first one-shot for the chapter "Units and Measurement" for the 2024-25 academic year.
• The speaker's name is Ashu Ghai, and he teaches at Science and Fun.
• The teaching approach involves both the heart and mind, focusing on understanding concepts.
• This year, one-shot videos will be presented in a more interesting format, with the first chapter lacking experiments but connecting to daily life.
• The new NCERT syllabus has removed topics like the parallax method and errors from the Units and Measurement chapter.

Quantities

• A quantity is defined as anything that can be measured.
• Examples of quantities include time, mass, and force.
• Love, hatred, and happiness are not considered quantities because they cannot be measured.

Units

• Units are words or symbols that are used to measure or derive quantities.
• Examples of units for mass are milligrams, grams, and kilograms.
• Examples of units for time are seconds, minutes, and hours.

Types of Units and Quantities

• There are two types of quantities: fundamental and derived.
• Fundamental units are assumed and have inherent stories behind them.
• There are seven fundamental units in physics.
• Derived units are made up of the seven fundamental units.

Seven Fundamental Units

• Mass: kilogram (kg)
• Time: second (s)
• Current: ampere (A)
• Length: meter (m)
• Amount of substance: mole (mol)
• Luminous intensity: candela (cd)
• Temperature: kelvin (K)

Derived Units Examples

• Force: Unit is the Newton (N), derived from mass and acceleration.
  • Formula: Force = Mass × Acceleration.
  • Unit: kg⋅m/s².
• Pressure: Unit is Pascal (Pa), derived from force and area.
  • Formula: Pressure = Force / Area.
  • Unit: kg/m⋅s².
• Numerous other units are derived from the seven fundamental units.

Dimensions

• Dimensions provide a symbolic representation of units, regardless of their specific unit.
• Unlike units, which can vary for the same quantity, dimensions are always the same.
• Mass is represented by the dimension [M], time by [T], and length by [L].
• The entire 11th-grade physics course will primarily use mass, length, and time dimensions.

Dimensions of Speed or Velocity

• Formula: Speed = Distance / Time.
• Dimension of Speed: [L]/[T] = [LT⁻¹].
• Standard way to represent dimension includes mass, length, and time.
• Alternate representation: [M⁰LT⁻¹].

Dimensions of Momentum

• Formula: Momentum = Mass × Velocity.
• Dimension of Momentum: [M][LT⁻¹] = [MLT⁻¹].

Dimensions of Kinetic Energy

• Formula: Kinetic Energy = 1/2 × Mass × Velocity².
• Dimension of Kinetic Energy: [M][LT⁻¹]² = [ML²T⁻²].
• Numerical constants do not have dimensions.
• Fact: All types of energy have the same dimensions, [ML²T⁻²].
• Energy and work have the same dimensions.

Dimensions of Acceleration

• Formula: Acceleration = Change in Velocity / Time.
• Dimension of Acceleration: [LT⁻¹]/[T] = [LT⁻²].

Dimensions of Force

• Formula: Force = Mass × Acceleration.
• Dimension of Force: [M][LT⁻²] = [MLT⁻²].

Dimensions of Torque

• Formula: Torque = Force × Distance.
• Dimension of Torque: [MLT⁻²][L] = [ML²T⁻²].
• Torque and energy are different things but they coincidentally have the same dimensions.

Dimensionless Quantities

• Example: Refractive Index
• Formula: Speed of Light in First Medium / Speed of Light in Second Medium.
• Since it is a ratio of two speeds, its dimension is [M⁰L⁰T⁰].

Principle of Homogeneity

• According to the principle of homogeneity, in an equation, each term must have the same unit and dimensions.
• Quantities can only be added or subtracted if they have the same dimensions.
• If x + y, then the unit of x and y will be the same, and dimension will also be the same.

Application of the Principle of Homogeneity

• It can be used to calculate the dimension of unknown quantities.
• For example: In an equation x + y t = a b, all three dimensions should be the same.
• Let's suppose x / y + √c=t² * x for the above equation x / y, √c, and t² * x all have the same dimensions.

Example Questions

• Calculate the dimensions of x given: Speed × Time + x = Area / Length.
  • Each terms dimension must be the same.
  • Dimension of x would be [L].
• Calculate dimension of x given: Force * Distance + x = Mass * Acceleration * Height
  • Dimension of x can be found by equating dimension of first term and is found to be [ML²T⁻²].

Applications of Dimensional Analysis

• To check if an equation is dimensionally correct.
• To convert the system of units.
• To derive a formula.

First Application - Checking Dimensional Correctness of Equations

• Example: 1/2 mv² = mgh
  • Equating the dimensions of LHS and RHS shows that equation is dimensionally correct.
• To prove equation actually correct, the value of dimensions of LHS and RHS need to be equal.
• The vice versa of above point is always true.
• Another important point is If equation is dimensionally NOT correct, then it cannot be correct.

Second Application - Conversion of System of Units

• Unit systems are categorized into three types MKS, CGS, and FPS, where MKS stands for Meter Kilogram Second, CGS for Centimeter Gram Second, and FPS Foot Pound Second respectively.
• It teaches how convert systems using dimension, such as converting MKS to CGS.

Conversion Formula

• The formula to convert units from one system to another is: N₂ = N₁ (M₁/M₂)^x (L₁/L₂)^y (T₁/T₂)^z, where
  • N₂ is the value in the new system,
  • N₁ is the value in the initial system,
  • M₁, L₁, and T₁ are the mass, length, and time units in initial system,
  • M₂, L₂, and T₂ are the mass, length, and time units in the new system,
  • x, y, and z are the dimensions regarding mass, length, and time.

Example Questions for Conversion of System of Unit

• Convert 7 kg²m²/s² to CGS unit where its dimension regarding M,L,T are found to be 2,2,-2.
  • This involves using the formula, converting kilograms to grams and meters to centimeters, and computing the new value in CGS unit.
  • Always identify starting and ending unit. Apply the formula carefully and cut like terms such as kg etc.

Third Application - Deriving Formula

• Explanation of why deriving formula is important in Physics.
• The power of mass, dimension and height are noted.
• Example using potential energy formula (PE = mgh), explaining each parameter and the derivation using dimensional analysis.

Example Question

• If time period of Pendulum (T) depends on Mass(M), Length(L) and Gravitational acceleration. Find the formula.
  • Time T ∝ Mass^x Length^y acceleration^z and we need to find x,y & z;
  • Find the combined dimension [M^x L^(y+z) T^-2z] for RHS;
  • Then the powers of all dimension should be equal regarding both LHS and RHS;
  • In the end formula for Time period = K √(l/g) comes out since power is L is 1/2 and for acceleration dimension is -1/2.

Limitations of this Method

• Cannot find value of all constants.
• cannot drive formula having having geometric functions.
• Cannot derive any more expressions containing more the one term.

Significant Figures

• Significant figures are important in scientific calculations and software development.
• First rule: all of non-zero digits are Significant. For example 342 has 3 significant values.
• Second rule if zero values exist in between two non-zero digits. Example 307 has also has three significant values.
• Third rule if the non-zero digits have leading zeros then values wont be called Significant. Example 0.00437 doesn't include values.
• Fourth and Fifth rule, if a non-zero ending is ended by trailing zero without decimal, then its insignificant. Values wont be included.But if decimal value is included it'll be significant.
• Sixth rule, any value with Scientific Notations like a*10^x then value a will contain Significant figures.

Rounding values with some special considerations.

• Rounding values are crucial in providing better result.
• First rule is that if any digit is followed number less than 5, then it won't be anything.
• Second rule is that if digit if follow by value greater than 5 will then increase by one.
• Third and most important if the digit is followed by value =5 then
  • if digit is even then do nothing
  • else increment value to make it even from odd.

Example for more explanation

• If multiplication of two digits contains multiple numbers of significant figures. Then Final answer would have number in Least number of Significant figures. Same rule will be applicable for division
• The final result of additions and subtractions should retain as many decimal places as there are in the number with the least decimal places.