Which Of The Following Are Dimensionally Consistent

Dimensional Consistency: A Deep Dive into Physical Equations

Determining which equations are dimensionally consistent is a fundamental concept in physics and engineering. It's a powerful tool for checking the validity of equations and ensuring that calculations are meaningful. This article will delve into the principles of dimensional analysis, explore techniques for checking dimensional consistency, and provide numerous examples to solidify your understanding. We'll cover the basics and then move into more complex scenarios, helping you master this essential skill.

Introduction: What is Dimensional Consistency?

Dimensional consistency, also known as dimensional homogeneity, refers to the principle that an equation describing a physical phenomenon must have the same dimensions on both sides. In simpler terms, the units of measurement on the left-hand side (LHS) of an equation must be equal to the units of measurement on the right-hand side (RHS). If this isn't the case, the equation is dimensionally inconsistent, indicating a likely error in its derivation or application. This is a crucial step in verifying the correctness of any physical formula. A dimensionally inconsistent equation cannot possibly be correct, although a dimensionally consistent equation isn't necessarily correct.

Fundamental Dimensions:

Before we delve into checking dimensional consistency, we need to understand fundamental dimensions. These are the basic building blocks upon which all other physical quantities are built. The most commonly used system is the International System of Units (SI), which uses seven fundamental dimensions:

• Length (L): Measured in meters (m)
• Mass (M): Measured in kilograms (kg)
• Time (T): Measured in seconds (s)
• Electric Current (I): Measured in amperes (A)
• Thermodynamic Temperature (Θ): Measured in Kelvin (K)
• Amount of Substance (N): Measured in moles (mol)
• Luminous Intensity (J): Measured in candelas (cd)

All other physical quantities can be expressed as combinations of these fundamental dimensions. For instance, speed is length divided by time (L/T), acceleration is length divided by time squared (L/T²), and force is mass times acceleration (ML/T²).

Techniques for Checking Dimensional Consistency:

To check if an equation is dimensionally consistent, follow these steps:

1. Identify the dimensions of each term: Express each term in the equation in terms of its fundamental dimensions (M, L, T, I, Θ, N, J). For example, velocity (v) has dimensions of L/T, acceleration (a) has dimensions of L/T², and force (F) has dimensions of ML/T².

2. Substitute the dimensions into the equation: Replace each physical quantity in the equation with its corresponding dimensions.

3. Simplify the equation: Use algebraic manipulation to simplify the equation, combining and canceling dimensions where possible.

4. Compare the dimensions on both sides: If the dimensions on the LHS are equal to the dimensions on the RHS, the equation is dimensionally consistent. If they are not equal, the equation is dimensionally inconsistent, suggesting an error.

Examples of Dimensionally Consistent and Inconsistent Equations:

Let's illustrate these steps with some examples:

Example 1: Newton's Second Law of Motion

Newton's second law states that F = ma (Force equals mass times acceleration).

1. Dimensions:

   • F (Force) = ML/T²
   • m (mass) = M
   • a (acceleration) = L/T²

2. Substitution: ML/T² = M * (L/T²)

3. Simplification: The equation simplifies to ML/T² = ML/T², which is clearly equal on both sides.

4. Conclusion: Newton's second law is dimensionally consistent.

Example 2: An Inconsistent Equation

Consider the equation: v = at² + x, where v is velocity, a is acceleration, t is time, and x is displacement.

1. Dimensions:

   • v (velocity) = L/T
   • a (acceleration) = L/T²
   • t (time) = T
   • x (displacement) = L

2. Substitution: L/T = (L/T²) * T² + L

3. Simplification: L/T = L + L = 2L. This simplifies to L/T = 2L.

4. Conclusion: The dimensions on the LHS (L/T) are not equal to the dimensions on the RHS (L), therefore this equation is dimensionally inconsistent. There is an error in this equation.

Example 3: Kinetic Energy

The equation for kinetic energy is KE = 1/2mv².

1. Dimensions:

   • KE (Kinetic Energy) = ML²/T² (Energy has dimensions of force times distance, which is (ML/T²) * L)
   • m (mass) = M
   • v (velocity) = L/T

2. Substitution: ML²/T² = M * (L/T)²

3. Simplification: ML²/T² = M * (L²/T²) = ML²/T²

4. Conclusion: The equation for kinetic energy is dimensionally consistent.

Example 4: A More Complex Example - Projectile Motion

Consider the equation for the horizontal range (R) of a projectile launched at an angle θ with initial velocity v₀:

R = (v₀²sin(2θ))/g

where:

• R = horizontal range (L)
• v₀ = initial velocity (L/T)
• θ = launch angle (dimensionless)
• g = acceleration due to gravity (L/T²)

1. Dimensions: R = L v₀² = (L/T)² = L²/T² sin(2θ) = dimensionless g = L/T²

2. Substitution: L = (L²/T²) * (1) / (L/T²)

3. Simplification: L = L²/T² * T²/L = L

4. Conclusion: The equation for the horizontal range of a projectile is dimensionally consistent.

Example 5: Involving Other Fundamental Dimensions - Electrical Power

Electrical power (P) is given by P = IV, where I is current and V is voltage.

1. Dimensions: P (Power) = ML²/T³ (Energy/Time) I (Current) = I (Amperes) V (Voltage) = ML²/T³I⁻¹ (Energy/Charge)

2. Substitution: ML²/T³ = I * ML²/T³I⁻¹

3. Simplification: ML²/T³ = ML²/T³

4. Conclusion: The equation for electrical power is dimensionally consistent.

Beyond Basic Checks: The Power of Dimensional Analysis

Dimensional analysis isn't just about verifying existing equations; it's a powerful tool for deriving equations and understanding relationships between physical quantities. If you know the relevant variables involved in a phenomenon, you can use dimensional analysis to determine the form of the equation relating them. This is particularly useful when the exact form of the equation is unknown or complex to derive through other means.

Frequently Asked Questions (FAQ):

• Q: If an equation is dimensionally consistent, does that guarantee it's correct?

  • A: No, dimensional consistency is a necessary but not a sufficient condition for the correctness of an equation. An equation can be dimensionally consistent but still contain errors in constants or functional forms.

• Q: What should I do if I find an equation is dimensionally inconsistent?

  • A: A dimensional inconsistency indicates a likely error. Carefully re-examine your derivation, check your units, and ensure that you've used the correct physical principles.

• Q: Can dimensionless quantities affect dimensional consistency?

  • A: No, dimensionless quantities (like angles or ratios) don't affect the dimensional consistency of an equation. They act as scaling factors without adding or removing any fundamental dimensions.

• Q: What are some common mistakes that lead to dimensional inconsistencies?

  • A: Common mistakes include incorrect unit conversions, misusing formulas, and neglecting fundamental constants in equations.

Conclusion:

Dimensional consistency is a critical aspect of working with physical equations. Mastering this concept significantly enhances your ability to analyze physical systems, verify the validity of equations, and prevent errors in calculations. By diligently following the steps outlined in this article, and practicing with various examples, you can develop a strong understanding of this fundamental concept and significantly improve the accuracy and reliability of your work in physics and engineering. Remember, while dimensional analysis is a powerful tool, it serves as a check for errors, not a guarantee of correctness. Always combine it with a thorough understanding of the underlying physical principles.