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Chapter 1: Problem 6

Consider the equation \(v=\frac{1}{3} z x t^{2}\) . The dimensions of the variables \(v, x,\) and \(t\) are \([\mathrm{L}] / \mathrm{T} ],[\mathrm{L}],\) and \([\mathrm{T}],\) respectively. The numerical factor 3 is dimensionless. What must be the dimensions of the variable \(z,\) such that both sides of the equation have the same dimensions? Show how you determined your answer.

Short Answer

Expert verified
The dimensions of \(z\) must be \([1/\text{T}^3]\) for dimensional consistency.

Step by step solution

01

Identify Given Dimensions

We are given the dimensions for the variables as follows: \(v = \frac{1}{3} z x t^2\), where \([v] = [\text{L/T}]\), \([x] = [\text{L}]\), and \([t] = [\text{T}]\). The equation must maintain dimensional consistency on both sides.
02

Determine Dimensions on the Left Side

The left side of the equation has the variable \(v\) with dimensions \([\text{L/T}]\). So, the left side dimensions are \([\text{L/T}]\).
03

Determine Dimensions on the Right Side

The right side of the equation is \(\frac{1}{3} z x t^2\). The dimensions for \(x\) are \([\text{L}]\) and for \(t^2\) are \([\text{T}^2]\). The dimensions for the right side become \([z][\text{L}][\text{T}^2]\).
04

Establish Dimensional Equality

Set the dimensions of both sides equal to each other to maintain dimensional consistency.\([\text{L/T}] = [z][\text{L}][\text{T}^2]\)
05

Solve for the Dimensions of \(z\)

We have the equation \([\text{L/T}] = [z][\text{L}][\text{T}^2]\). To solve for \([z]\), cancel out \([\text{L}]\) from both sides:\([1/\text{T}] = [z][\text{T}^2]\).Now, solve for \([z]\):\([z] = [1/\text{T}^3]\).
06

Verify the Solution

Substituting \([z] = [1/\text{T}^3]\) back into the right side, we get \([1/\text{T}^3][\text{L}][\text{T}^2] = [\text{L/T}]\), which matches the left side dimensions. This confirms the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics Equations
In the world of physics, equations serve as mathematical representations of physical phenomena. They help us understand the relationships between different quantities. Physics equations are structured around variables that typically represent various physical quantities with their specific dimensions. In any physics equation, it is essential that both sides of the equation have equivalent dimensional units. This concept is known as dimensional consistency and ensures that we are comparing apples to apples. Physics equations often include constants and variables that interact to describe real-world situations. These equations can express relationships such as motion, force, energy, and other physical interactions. Understanding the dimensions and units each variable carries can aid in verifying the correctness of an equation, as well as in problem-solving exercises where such concepts are crucial.
Unit Consistency
Ensuring unit consistency, also known as dimensional consistency, is a critical part of working with physics equations. This means that both sides of an equation must result in the same type of dimensional units, allowing for valid comparisons and calculations. It helps prevent mathematical errors and incorrect interpretations of physical laws. For instance, in our exercise, we needed to ensure both sides of the equation had the same dimensions. On the left, we had a velocity, \([\text{L/T}]\), and on the right, the challenge was to figure out the appropriate dimensions for the variable \(z\). Ensuring the proper dimensions for \(z\), \[z = [1/\text{T}^3]\], allowed us to achieve this unit consistency, demonstrating how careful analysis helps maintain the accuracy of physics computations. Without unit consistency, equations would lead to nonsensical or incorrect results, highlighting the importance of this concept.
Problem-Solving Steps
Problem-solving in physics often involves a series of logical steps that guide us to a solution. This structured approach helps in dissecting complicated problems and systematically finding solutions. Step-by-Step Method:
  • Identify Given Dimensions: Start by understanding the dimensions you are working with. This forms the basis of maintaining dimensional consistency.
  • Analyze Each Side: Break down the equation to determine the dimensional units on both sides. Ensuring each component aligns dimensionally with the established understanding is key.
  • Determine Missing Dimensions: If a dimension is unknown, deduce it by ensuring both sides of the equation achieve dimensional parity. This often involves basic algebraic manipulations.
  • Verify Consistency: Once you have assigned dimensions, go back to affirm that the dimensions indeed match on both sides, reinforcing the solution's validity.
By following these structured steps, the complex process becomes manageable, leading to a more confident and accurate solution.

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Most popular questions from this chapter

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