Calculating Mass From Density And Volume A Physics Problem

Jul 15, 2025 by ADMIN 59 views

Understanding Density and Mass Calculation for an Aluminum Cylinder

In the realm of physics, understanding the relationship between density, mass, and volume is crucial. Density, defined as mass per unit volume, is an intrinsic property of a substance. This means that a substance's density remains constant regardless of the amount of the substance present. The formula that connects these three concepts is quite straightforward: Density = Mass / Volume. This article will delve into calculating the mass of an aluminum cylinder given its density and volume, providing a step-by-step approach to solving the problem and a comprehensive understanding of the underlying principles.

The problem at hand involves an aluminum cylinder. Aluminum is a widely used metal known for its lightweight and high strength-to-weight ratio, making it ideal for various applications ranging from aerospace to everyday household items. The density of aluminum is given as $2.70 imes 10^3$ kilograms per cubic meter $(kg / m^3)$ . This value tells us how much mass is contained in each cubic meter of aluminum. We are also given that the aluminum cylinder has a volume of $1.50 m^3$ . The objective is to calculate the mass of this cylinder. This is a practical application of the density formula, where we use the given density and volume to find the mass. Understanding how to manipulate and apply this formula is fundamental in physics and engineering.

To find the mass of the aluminum cylinder, we will use the formula that relates density, mass, and volume: Density = Mass / Volume. We are given the density of aluminum as $2.70 imes 10^3 kg/m^3$ and the volume of the cylinder as $1.50 m^3$ . Our goal is to find the mass. We can rearrange the formula to solve for mass:

Mass = Density × Volume

Now, we can substitute the given values into the formula:

Mass = $(2.70 imes 10^3 kg/m^3) imes (1.50 m^3)$

Performing the multiplication:

Mass = $2.70 imes 1.50 imes 10^3 kg$

Mass = $4.05 imes 10^3 kg$

Therefore, the mass of the aluminum cylinder is $4.05 imes 10^3 kg$ . However, upon reviewing the provided options, we notice that this answer is not directly listed. It is crucial to double-check our calculations and the given information to ensure accuracy. Let’s re-examine the multiplication:

$2.70 imes 1.50 = 4.05$

So, the mass is indeed $4.05 imes 10^3 kg$ . It seems there might be a discrepancy between our calculated answer and the provided options. It's possible that there was a mistake in the original options, or the question might have been misinterpreted. In such cases, it's important to rely on the correct calculation and, if necessary, highlight the discrepancy.

If we consider the closest answer among the options, option C, $1.80 imes 10^3 kg$ , is significantly different from our calculated value of $4.05 imes 10^3 kg$ . This reinforces the likelihood of an error in the provided options or a misunderstanding of the question. Therefore, the most accurate answer based on our calculations is $4.05 imes 10^3 kg$ , which is not among the given choices.

Identify the Given Information:
- Density of aluminum ( $ρ$ ) = $2.70 imes 10^3 kg/m^3$
- Volume of the cylinder ( $V$ ) = $1.50 m^3$
Recall the Density Formula:
- Density = Mass / Volume ( $ρ = rac{m}{V}$ )
Rearrange the Formula to Solve for Mass:
- Mass = Density × Volume ( $m = ρ imes V$ )
Substitute the Given Values into the Formula:
- $m = (2.70 imes 10^3 kg/m^3) imes (1.50 m^3)$
Perform the Multiplication:
- $m = 4.05 imes 10^3 kg$

Now, let's analyze the given options in the context of our calculated mass:

A. $5.56 imes 10^{-4} kg$ : This value is extremely small compared to our calculated mass of $4.05 imes 10^3 kg$ . It is several orders of magnitude smaller, indicating a significant discrepancy.

B. $1.50 imes 10^{-3} kg$ : This option is also significantly smaller than our calculated mass. It is three orders of magnitude less than $1 kg$ , whereas our result is in the thousands of kilograms.

C. $1.80 imes 10^3 kg$ : This option is closer to our calculated value but still substantially different. Our calculated mass is more than twice this value, indicating a considerable difference.

Given these comparisons, it is evident that none of the provided options accurately represent the mass of the aluminum cylinder based on the given density and volume. The closest option, C, is less than half of our calculated value, further highlighting the discrepancy.

Understanding the fundamental concepts of density, mass, and volume is essential in physics. These concepts are interconnected and play a crucial role in various scientific and engineering applications. Density, as previously mentioned, is defined as mass per unit volume. It is an intrinsic property of a substance, meaning it does not depend on the amount of substance present. For example, the density of pure water is approximately $1000 kg/m^3$ at standard conditions, whether you have a cup of water or an entire lake.

Mass, on the other hand, is a measure of the amount of matter in an object. It is a fundamental property of an object and is measured in kilograms (kg) in the International System of Units (SI). Mass is different from weight, which is the force exerted on an object due to gravity. An object's mass remains constant regardless of its location, while its weight can vary depending on the gravitational field.

Volume is the amount of space that an object occupies. It is measured in cubic meters ( $m^3$ ) in the SI system. The volume of a regular-shaped object, such as a cylinder, can be calculated using specific formulas. For a cylinder, the volume is given by the formula $V = πr^2h$ , where $r$ is the radius of the base and $h$ is the height of the cylinder. However, in this problem, the volume was provided directly, simplifying the calculation.

The relationship between these concepts is expressed by the formula: Density = Mass / Volume. This formula can be rearranged to solve for any of the three variables if the other two are known. This is a fundamental tool in physics and is used in various applications, such as determining the purity of a substance, calculating the buoyancy of an object in a fluid, and designing structures and machines.

The concepts of density, mass, and volume have numerous practical applications and real-world examples. Understanding these concepts is crucial in fields such as engineering, materials science, and even cooking.

In engineering, density is a critical factor in designing structures and machines. For example, in aerospace engineering, the density of materials used in aircraft construction is a significant consideration. Aluminum alloys, known for their low density and high strength, are commonly used to reduce the weight of aircraft, which improves fuel efficiency and performance. Similarly, in civil engineering, the density of concrete and steel is essential for designing stable and durable buildings and bridges.

In materials science, density is used to characterize and identify different materials. Each substance has a unique density, which can be used as a fingerprint to identify it. This is particularly useful in quality control and material testing. For example, the density of a metal can indicate its purity and composition. Additionally, density plays a crucial role in the development of new materials with specific properties, such as lightweight composites and high-density alloys.

In everyday life, the concept of density is used in various applications. For instance, when cooking, understanding the density of ingredients can help in measuring and mixing them correctly. In maritime activities, density is crucial for understanding buoyancy and the stability of ships. Ships are designed to have an overall density less than that of water, allowing them to float. The principles of density are also applied in the design of submarines, which can control their buoyancy by adjusting their density.

When working with density, mass, and volume calculations, several common mistakes can occur. Being aware of these pitfalls and knowing how to avoid them can significantly improve accuracy and understanding.

Incorrect Unit Conversions: One of the most common mistakes is using inconsistent units. Density is often given in $kg/m^3$ , while volume might be given in cubic centimeters ( $cm^3$ ) or liters (L). It is crucial to convert all quantities to the same units before performing calculations. For example, to convert from $cm^3$ to $m^3$ , divide by $10^6$ (since $1 m = 100 cm$ , so $1 m^3 = (100 cm)^3 = 10^6 cm^3$ ).
Misunderstanding the Formula: Another common mistake is misinterpreting or misremembering the density formula. It is essential to understand that Density = Mass / Volume, and to rearrange it correctly when solving for mass or volume. Writing down the formula before substituting values can help prevent errors.
Calculation Errors: Simple arithmetic errors can lead to incorrect results. It is always a good practice to double-check calculations, especially when dealing with scientific notation. Using a calculator can help reduce the likelihood of errors.
Ignoring Significant Figures: In scientific calculations, it is important to pay attention to significant figures. The final answer should be rounded to the same number of significant figures as the least precise measurement. Ignoring significant figures can lead to inaccuracies and misinterpretations of the results.
Not Understanding the Concepts: A lack of conceptual understanding can lead to errors in problem-solving. It is crucial to understand the definitions of density, mass, and volume and how they relate to each other. This understanding will help in choosing the correct approach and interpreting the results correctly.

To avoid these mistakes, it is recommended to:

Always write down the formula before substituting values.
Pay attention to units and convert them if necessary.
Double-check calculations and use a calculator.
Consider significant figures in the final answer.
Ensure a thorough understanding of the underlying concepts.

In summary, this article has provided a comprehensive guide to calculating the mass of an aluminum cylinder given its density and volume. We have demonstrated the step-by-step solution, discussed the key concepts of density, mass, and volume, explored practical applications and real-world examples, and highlighted common mistakes to avoid. The calculated mass of the aluminum cylinder is $4.05 imes 10^3 kg$ , which does not match any of the provided options, suggesting a potential error in the options. By understanding the fundamental principles and following a systematic approach, you can confidently solve similar problems and apply these concepts in various scientific and engineering contexts. The relationship between density, mass, and volume is a cornerstone of physics, and mastering it is crucial for a deeper understanding of the physical world.