Dimensional Homogeneity: Unveiling √xy's Mystery In Physics

Hey guys! Today, we're diving into a cool physics problem that deals with dimensional homogeneity. Don't worry if you're not a physics whiz – we'll break it down step by step. The question is: If the following expression is dimensionally homogeneous, what physical quantity does it represent: √xy. QRX = 5W Cos TC 2 RY? Where Q represents power, R represents velocity, W represents work, and we need to figure out what the square root of xy corresponds to. Let's get started, shall we?

Understanding Dimensional Homogeneity and the Given Expression

So, what's dimensional homogeneity all about? Basically, it means that every term in an equation must have the same dimensions. Think of it like this: you can't add apples and oranges. In physics, you can't add quantities with different units unless you convert them into the same units. This principle helps us check if our equations make sense. This is essential for ensuring the validity of physics equations. The equation √xy. QRX = 5W Cos TC 2 RY is given. It seems a bit complicated, but we'll break it down. The goal is to determine the physical quantity represented by √xy. To achieve this, we'll use the concept of dimensional analysis, a powerful tool to check the consistency of physical equations and to determine the nature of unknown quantities.

First, let's decode the terms: Q represents Power, R is Velocity, and W is Work. What about the rest of the equation? x and y are unknown quantities, and our task is to identify what they represent in the equation. Cos TC 2 RY is a bit of a trick. The cosine function is dimensionless – it doesn't have any units. The number 5 is also dimensionless. Therefore, the only terms that contribute to the dimensions of the equation are √xy, Q, R, and W.

Now that we understand the basics, let's move on to the next step: analyzing the dimensions of each term. Remember, the whole equation must be dimensionally consistent. If we find a combination of units that balance the equation, we'll have our answer. We will do this using a step-by-step approach. It's all about balancing the units on both sides. Keep in mind that the dimensions must match on both sides of the equation.

Dimensional Analysis: Unveiling the Dimensions

Alright, let's get our hands dirty with some dimensional analysis. We need to find the dimensions of each term in the equation. It's like a treasure hunt, but instead of gold, we're looking for the fundamental units: mass (M), length (L), and time (T). Let's start with the knowns:

• Power (Q): Power is the rate at which work is done, so its dimensions are [ML²T⁻³].
• Velocity (R): Velocity is distance over time, giving us dimensions of [LT⁻¹].
• Work (W): Work is force times distance, so its dimensions are [ML²T⁻²].

Now, let's rewrite the given equation and represent it in terms of dimensions:

√xy . [ML²T⁻³] . [LT⁻¹] = [ML²T⁻²]

To make things clearer, we'll simplify it:

√xy . [M L³ T⁻⁴] = [M L² T⁻²]

Now, we need to find the dimensions of √xy. To do this, we'll isolate √xy by dividing both sides of the equation by [M L³ T⁻⁴]:

√xy = [M L² T⁻²] / [M L³ T⁻⁴]

When dividing exponents, we subtract them. So, let's simplify this:

√xy = [M⁰ L⁻¹ T²]

Here, M⁰ means that mass is not present in the dimensions of √xy. So, the final dimensions are:

√xy = [L⁻¹ T²]

Now, we have the dimensions of √xy. We’ll need to figure out what physical quantity has these dimensions.

Let’s find out how to get to the solution. We've successfully broken down the problem and found the dimensions of the unknown quantity, √xy. Using dimensional analysis, we've determined that the dimensions of √xy are [L⁻¹ T²]. The units are inverse length multiplied by time squared. Based on the options presented, we can deduce the correct answer by relating these dimensions to common physical quantities. This step involves matching the derived dimensions with those of known physical quantities to identify the correct answer. The goal is to link the calculated dimensions to a well-known physical quantity.

Identifying the Physical Quantity and The Correct Answer

Okay, guys, we're in the final stretch! We have the dimensions of √xy which are [L⁻¹ T²]. Now, let's check the options and see which one matches these dimensions.

• (a) Velocity: Velocity has dimensions of [LT⁻¹].
• (b) Force: Force has dimensions of [MLT⁻²].
• (c) Power: Power has dimensions of [ML²T⁻³].
• (d) Work: Work has dimensions of [ML²T⁻²].
• (e) √mass: Mass has dimensions of [M], and we are looking at the square root of mass. So it wouldn’t fit with the units we have found.

None of the standard physical quantities directly matches the dimensions [L⁻¹ T²]. There seems to be an error in the initial options, or the question might require a deeper understanding beyond basic dimensions. Because of that, we can't find the correct answer in the options provided. However, based on the analysis, there is no direct correlation with the given options. We are left with an inconsistency. The result shows a mismatch between the calculated dimensions and the provided options, indicating either an error in the question, the options, or a more complex relationship that is not immediately apparent with a basic dimensional analysis. This is a great example of how dimensional analysis can help us spot errors or inconsistencies in physics problems.

So, the answer is not directly present in the options, and the analysis shows the dimensions [L⁻¹ T²] do not correspond with any of the choices. This indicates a possible error in the question or the options provided. It's important to be aware of these potential issues while solving physics problems. Sometimes, the provided information might be incomplete or incorrect. Remember, the key is to apply your knowledge and use dimensional analysis to check the validity of the equations.

Conclusion: The Power of Dimensional Analysis

In this journey, we've used dimensional analysis to understand the dimensions of the unknown quantity √xy. Although we couldn't find the exact answer in the options, we've learned how to use this technique to check the consistency of equations and identify potential errors.

Remember, dimensional analysis is a handy tool for any physics student. It helps you check your work, understand the relationships between physical quantities, and even spot mistakes in problems.

Keep practicing, and you'll become a pro at it in no time! Thanks for sticking around, and keep exploring the amazing world of physics!