Calculate The Length Of The Side Of The Square Brass Plate In Centimeters, Given Its Thickness Is 2mm, Mass Is 1.05kg, And Density Is 8.4g/cm³.

Introduction

In this article, we will delve into a classic physics problem involving the calculation of the side length of a square brass plate. This problem elegantly combines concepts of density, mass, and volume, offering a practical application of fundamental physics principles. Understanding these concepts is crucial not only for academic success in physics but also for appreciating the physical world around us. This comprehensive guide will walk you through the problem-solving process step-by-step, ensuring you grasp the underlying logic and can apply it to similar scenarios. We will also discuss the importance of unit conversions in physics calculations and how they can significantly impact the final result. The problem we will address is a perfect example of how seemingly simple measurements like mass and thickness can be used to determine other physical dimensions, such as the length of a side, when combined with material properties like density.

Problem Statement

We are given a square brass plate with a thickness of 2mm and a mass of 1.05kg. The density of the brass is known to be 8.4g/cm³. Our objective is to calculate the length of a side of the square plate in centimeters. This problem requires us to utilize the relationship between density, mass, and volume, and to apply geometric principles to relate the volume to the side length of the square plate. Before we jump into the calculations, it's important to understand each of these concepts individually. Density is a fundamental property of matter that describes how much mass is contained in a given volume. It's a crucial concept in many areas of physics and engineering, from material science to fluid mechanics. Mass, on the other hand, is a measure of the amount of matter in an object. It's a fundamental property that resists acceleration and is often confused with weight, which is the force of gravity acting on mass. Volume is the amount of three-dimensional space occupied by an object. Understanding these definitions is the first step towards solving the problem.

Key Concepts: Density, Mass, and Volume

Before diving into the calculations, let's solidify our understanding of the core concepts involved: density, mass, and volume. Density, often represented by the Greek letter ρ (rho), is a material's mass per unit volume. Mathematically, it's expressed as: ρ = m/V, where 'm' is mass and 'V' is volume. This equation tells us that a denser material has more mass packed into the same amount of space. Brass, for example, is denser than aluminum, meaning a brass object will be heavier than an aluminum object of the same size. Mass, as we've touched upon, is a measure of the amount of matter in an object. It's an intrinsic property that doesn't change regardless of location or gravitational force. The standard unit of mass in the International System of Units (SI) is the kilogram (kg). In our problem, the mass of the brass plate is given in kilograms, which we may need to convert depending on the units used for other quantities. Finally, volume is the three-dimensional space occupied by an object. For a rectangular object like our brass plate, the volume can be calculated by multiplying its length, width, and height (or thickness in this case). The standard unit of volume in the SI system is the cubic meter (m³), but we'll often encounter it in liters (L) or cubic centimeters (cm³), especially in problems involving smaller objects. In the context of our square brass plate, the volume will be related to the square of the side length multiplied by the thickness. Understanding these relationships and how they interact is key to solving the problem at hand.

Step-by-Step Solution

Now, let's break down the solution into manageable steps.

1. Convert Units

As with most physics problems, the first crucial step is to ensure that all quantities are expressed in consistent units. We are given the thickness in millimeters (mm), the mass in kilograms (kg), and the density in grams per cubic centimeter (g/cm³). To maintain consistency, let's convert everything to centimeters and grams. We know that 1 cm = 10 mm, so the thickness of the plate is 2 mm / 10 mm/cm = 0.2 cm. Next, we need to convert the mass from kilograms to grams. Since 1 kg = 1000 g, the mass of the plate is 1.05 kg * 1000 g/kg = 1050 g. By performing these conversions upfront, we avoid potential errors later in the calculations. Unit conversions are a fundamental aspect of physics problem-solving, and mastering them is essential for accuracy. It's always a good practice to double-check your conversions and ensure they make logical sense. In this case, converting to centimeters and grams allows us to work with units that are directly compatible with the given density, simplifying the subsequent calculations.

2. Calculate the Volume

The next step is to calculate the volume of the brass plate using the relationship between density, mass, and volume. As we discussed earlier, the formula for density is ρ = m/V. We can rearrange this formula to solve for volume: V = m/ρ. We now have the mass (m = 1050 g) and the density (ρ = 8.4 g/cm³) in consistent units. Plugging these values into the formula, we get: V = 1050 g / 8.4 g/cm³ = 125 cm³. This volume represents the total amount of space occupied by the brass plate. It's important to note that the units cancel out correctly, leaving us with the volume in cubic centimeters, which is what we expect. This calculation demonstrates the power of the density formula in relating macroscopic properties like mass and volume. By knowing the density of a material, we can determine its volume if we know its mass, or vice versa. This is a crucial tool in many scientific and engineering applications, such as determining the purity of a substance or calculating the amount of material needed for a specific construction project.

3. Determine the Side Length

Now that we have the volume of the plate, we can determine the side length. Since the plate is square, its volume can be expressed as V = side_length * side_length * thickness, or V = s² * t, where 's' is the side length and 't' is the thickness. We know the volume (V = 125 cm³) and the thickness (t = 0.2 cm), so we can solve for the side length 's'. Rearranging the formula, we get: s² = V / t. Plugging in the values, we have: s² = 125 cm³ / 0.2 cm = 625 cm². To find the side length 's', we take the square root of both sides: s = √625 cm² = 25 cm. Therefore, the length of one side of the square brass plate is 25 cm. This final calculation elegantly ties together all the previous steps, demonstrating how the initial information about mass, density, and thickness can be used to determine a linear dimension of the object. It also highlights the importance of understanding geometric relationships in solving physics problems. The fact that the plate is square allows us to relate its volume directly to the square of its side length, making the calculation straightforward once we have the volume.

Answer

The length of the side of the square brass plate is 25 cm.

Conclusion

This problem demonstrates a practical application of the concepts of density, mass, and volume. By carefully converting units, applying the density formula, and utilizing geometric relationships, we successfully calculated the side length of the square brass plate. This type of problem-solving approach is fundamental in physics and other scientific disciplines. It reinforces the importance of understanding the relationships between physical quantities and the need for accurate unit conversions. Beyond the specific solution, this exercise provides valuable insights into how physical properties can be used to determine geometric dimensions, a principle that is widely applied in engineering, material science, and many other fields. The ability to break down complex problems into smaller, manageable steps, as we did here, is a critical skill for any aspiring scientist or engineer. By mastering these fundamental concepts and problem-solving techniques, you will be well-equipped to tackle more challenging physics problems in the future.

This article has hopefully provided a clear and comprehensive understanding of how to calculate the side length of a square brass plate given its mass, thickness, and density. By understanding the underlying principles and applying a systematic approach, you can confidently solve similar problems and deepen your understanding of physics.