Dimensional Analysis & Its Calculations | Chemistry for EmSAT Achieve PDF Download

Understanding Dimensional Analysis and Physical Quantities

Dimensional analysis is a fascinating concept that helps us answer intriguing questions about the nature of different physical quantities. It allows us to compare and relate quantities that may seem dissimilar at first glance. Let's delve into this topic to grasp its significance.

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Dimensional Analysis

Dimensional analysis involves analyzing the dimensions and dimensional formulas of physical quantities to establish connections between them. This method relies on fundamental principles:

Browse more Topics under Units and Measurement

• The International System of Units
• Measurement of Length, Mass, and Time
• Significant Figures
• Accuracy, Precision of Instruments, and Errors in Measurement

The Concept of Physical Laws

Physical laws remain consistent regardless of the units used to measure a quantity. If we denote the measured value of a physical quantity in one unit system as \( n_1 a_1 \) and in another system as \( n_2 a_2 \), they must be equal due to the nature of physical laws.

This relationship can be represented as: \( n_1 a_1 = n_2 a_2 \) (Equation 1)

The Principle of Homogeneity

The principle of homogeneity states that the dimensions on both sides of an equation must be consistent throughout. If the dimensions differ, the equation loses its physical meaning. For instance,...

Dimensional Analysis

• Principle of Homogeneity

  When we have two sets of quantities with the same dimensions, we can equate their powers. For instance, if [M]^{a} in one system is equivalent to [M]^{x} in another, then we can say that a = x. The same applies to dimensions b and y, as well as c and z.

• Applications of Dimensional Analysis

  • Conversion of Units

    Regardless of the system of units used, the dimensions of a physical quantity remain constant. If we have fundamental quantities M_1, L_1, T_1 in one system and M_2, L_2, T_2 in another, we can express a quantity Q as:

    In the first system: Q = n_1u_1 = n_1[M_1^aL_1^bT_1^c]

    In the second system: Q = n_2u_2 = n_2[M_2^aL_2^bT_2^c]

    By equating these two expressions, we can determine the values of n_1 or n_2.

  • Checking Equation Consistency

    All physical equations must exhibit consistency in their dimensions. For example, if the dimensions on the left-hand side do not match those on the right-hand side, the equation is not consistent. An inconsistent equation could be: F = m^2 × a.

Physical Phenomenon and Relations between Physical Quantities

Finding Relationships in Physical Phenomena

The principle of Homogeneity helps us establish connections between different physical quantities in a given phenomenon. Let's understand this through an example.

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Solved Example: Time Period of a Simple Pendulum

Consider a simple pendulum where the time period (T) depends on:

• Length of the pendulum (L),
• Mass of the bob (m),
• Acceleration due to gravity (g).

By observation, we can express T ∝ L^αm^βg^γ or ... (5)

[T] = [L]^α[M]^β[L]^γ[T]^-2γ

Solving for α, β, and γ, we get:

α = 0; β = 0; γ = -1/2

Substituting these values back into equation (5), we find:

T ∝ √(L/g)

Example: Converting 1 Joule to Erg

Joule is the SI unit of work, while erg is the unit in the CGS system. Let's denote them as the first and second systems of units, respectively. Given 1J = 1, we aim to find n_2.

From the equation (4):

[M_1^aL_1^bT_1^c] = n_2[M_2^aL_2^bT_2^c]

The dimensional formula of work is [M^1L^2T^-2], which gives us a = 1, b = 2, and c = -2.

Summary of Key Concepts

Equation Explanation

• When two quantities are equal, their dimensions are also equal: [M^1L^2T^-2] = n2[M^1L^2T^-2]
• For instance, if M1 = 1 kg, L1 = 1 m, and T1 = 1 s, and M2 = 1 g, L2 = 1 cm, T2 = 1 s
• We can find n2 by dividing the dimensions: n2 = [M1^1L1^2T1^-2] / [M2^1L2^2T2^-2]

Calculation of n2

• To calculate n2 for T1^-2, we use specific values: [1kg^1*1g^2*1s^-2] / [1g^1*1cm^2*1s^-2]
• This results in n2 = 10^7

Energy Conversion

• Hence, 1 Joule (J) is equivalent to 10^7 erg

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Units and Measurement

• Standard Measurement Units
• Unit of Vibration
• SI Units List
• Units of Measurement
• Unit of Weight
• Geiger Counter
• Mass And Weight
• Time Measurement
• Precision Definition and Examples
• How to Calculate Cubic Meters?

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Units and Measurement

• Standard Measurement Units: These are established units used for measuring quantities such as length, mass, time, and more. For example, meters for length and kilograms for mass.
• Unit of Vibration: Refers to the specific unit used to quantify vibrations, often measured in hertz (Hz). For instance, the vibration frequency of a guitar string.
• SI Units List: The International System of Units (SI) provides a standard set of units for measurements worldwide, including meters, kilograms, seconds, and others.
• Units of Measurement: Various units are employed to quantify different properties, like liters for volume and Newtons for force.
• Unit of Weight: Denotes the unit used to measure the heaviness of an object, typically in kilograms or pounds.
• Geiger Counter: A device utilized for detecting and measuring ionizing radiation, commonly used in nuclear physics and medicine.
• Mass And Weight: While mass signifies the amount of matter in an object, weight is the force exerted on it due to gravity. For example, a person's mass remains constant, but their weight varies depending on gravitational pull.
• Time Measurement: Time is measured in units such as seconds, minutes, and hours, essential for tracking events and processes.
• Precision Definition and Examples: Precision refers to the degree of exactness in a measurement. For instance, a ruler marked in millimeters offers greater precision compared to one marked in centimeters.
• How to Calculate Cubic Meters? Cubic meters measure volume and are calculated by multiplying length, width, and height in meters. For example, to find the volume of a box, you multiply its three dimensions.

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