Calculating Mass From Gravitational Potential Energy A Physics Problem Solved

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Introduction

In the realm of physics, understanding the relationship between gravitational potential energy, mass, height, and gravitational acceleration is crucial. This article delves into a classic physics problem, where we aim to determine the mass of a ball given its gravitational potential energy and height. We will explore the fundamental concepts, the formula involved, and the step-by-step solution to this problem. By the end of this exploration, you'll not only have the answer but also a solid grasp of the underlying principles. In this particular scenario, we are presented with a ball possessing 116.62 J of gravitational potential energy while positioned at a height of 85 meters. Our objective is to calculate the mass of this ball, utilizing the principles of gravitational potential energy. The concept of gravitational potential energy is pivotal in physics, particularly in mechanics. It quantifies the energy an object possesses due to its position in a gravitational field. Several factors influence this energy, including the object's mass, its height above a reference point, and the gravitational acceleration acting upon it. To unravel the mass of the ball in question, we will employ the well-established formula for gravitational potential energy, which mathematically links these factors. This formula not only provides a means to calculate potential energy but also allows us to determine any one of the variables if the others are known. In our case, we know the gravitational potential energy and the height, and we are tasked with finding the mass. This involves rearranging the formula to isolate mass as the subject, a common practice in physics problem-solving. This article will walk you through each step, ensuring a clear understanding of the process and the underlying physics concepts. By understanding the relationship between gravitational potential energy, mass, and height, we can accurately determine the mass of an object given its potential energy and height within a gravitational field. This principle finds applications in diverse fields, ranging from engineering to astrophysics. Throughout our discussion, we'll emphasize the importance of accurate unit conversions and the proper application of the formula to ensure precise results. The problem at hand serves as a practical example of how theoretical physics concepts translate into real-world calculations. By working through this problem, we gain a deeper appreciation for the interconnectedness of physical quantities and the power of mathematical formulas in describing the world around us. So, let's embark on this journey of discovery and unveil the mass of the ball, step by step.

Understanding Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. The higher an object is, the more GPE it has. This energy has the potential to be converted into other forms of energy, such as kinetic energy when the object falls. The main factors influencing gravitational potential energy include the mass of the object (m), the height of the object above a reference point (h), and the acceleration due to gravity (g), which is approximately 9.8 m/s² on Earth. The relationship between these factors is expressed by the formula: GPE = mgh. This equation is the cornerstone of our calculation, allowing us to quantify the energy stored in an object due to its position within a gravitational field. The understanding of gravitational potential energy is not just a theoretical concept; it has practical implications in various fields. For instance, in engineering, it is crucial in designing structures and systems involving height and potential energy conversion, such as roller coasters or hydroelectric dams. In everyday life, we experience GPE constantly, from holding an object in the air to understanding how objects fall. The formula GPE = mgh is a powerful tool for analyzing these situations. It allows us to predict how much energy an object has at a certain height and how much energy it will release if it falls. Moreover, this concept extends beyond simple scenarios. In astrophysics, for example, GPE plays a significant role in understanding the dynamics of celestial bodies, such as the formation of stars and the movement of planets. The gravitational potential energy of a star system influences its stability and evolution. Similarly, in satellite mechanics, GPE is a critical factor in determining the orbits of satellites around the Earth. Understanding the interplay between GPE and kinetic energy is essential for predicting satellite trajectories and ensuring their proper functioning. The principle of conservation of energy dictates that GPE can be transformed into other forms of energy, and vice versa. When an object falls, its GPE is converted into kinetic energy, the energy of motion. This conversion is a fundamental aspect of physics and is governed by the laws of thermodynamics. In summary, gravitational potential energy is a fundamental concept in physics with wide-ranging applications. It is the energy an object possesses due to its position in a gravitational field and is determined by its mass, height, and the acceleration due to gravity. Understanding GPE is essential for analyzing a variety of physical phenomena, from the motion of everyday objects to the dynamics of celestial bodies.

Formula for Gravitational Potential Energy

The formula for gravitational potential energy (GPE) is a cornerstone in solving problems related to energy and motion in physics. It mathematically connects the potential energy of an object to its mass, height, and the gravitational acceleration. The formula is expressed as: GPE = mgh, where:

  • GPE represents the gravitational potential energy, measured in Joules (J).
  • m is the mass of the object, measured in kilograms (kg).
  • g is the acceleration due to gravity, which is approximately 9.8 m/s² on Earth.
  • h is the height of the object above a reference point, measured in meters (m).

This formula reveals that the gravitational potential energy is directly proportional to the mass of the object, its height, and the gravitational acceleration. This means that if any of these factors increase, the GPE will also increase proportionally. Understanding this relationship is crucial for solving problems involving GPE. The formula GPE = mgh is not just a mathematical equation; it is a physical representation of how energy is stored in an object due to its position in a gravitational field. The higher the object, the more potential it has to do work if it is allowed to fall. This potential is quantified by the GPE. The gravitational acceleration (g), being a constant value on Earth (approximately 9.8 m/s²), plays a significant role in determining the GPE. It represents the force that gravity exerts on the object, pulling it towards the Earth. The mass (m) of the object is another crucial factor. A heavier object will have more GPE at the same height compared to a lighter object because it requires more energy to lift the heavier object against gravity. The height (h) is the vertical distance of the object from a reference point, often the ground. The higher the object is lifted, the more work is done against gravity, and the more GPE it gains. This height is crucial because it defines the object's position within the gravitational field. Applying this formula correctly requires careful attention to units. The mass must be in kilograms, the height in meters, and the gravitational acceleration in meters per second squared to obtain the GPE in Joules. Consistency in units is essential for accurate calculations in physics. Moreover, the formula can be rearranged to solve for any of the variables if the others are known. For example, if we know the GPE, mass, and gravitational acceleration, we can solve for the height. This flexibility makes the formula a versatile tool in problem-solving. In our specific problem, we are given the GPE and the height, and we need to find the mass. This requires rearranging the formula to isolate mass as the subject. This process of rearranging formulas is a common practice in physics and is a valuable skill for any aspiring physicist or engineer. In summary, the formula GPE = mgh is a fundamental equation in physics that relates gravitational potential energy to mass, height, and gravitational acceleration. Understanding this formula and its applications is essential for solving problems involving energy and motion in gravitational fields. It is a powerful tool that allows us to quantify the potential energy stored in an object due to its position and to make predictions about its behavior.

Step-by-Step Solution

Now, let's apply our understanding of gravitational potential energy to solve the given problem step-by-step. We are given that the ball has a gravitational potential energy (GPE) of 116.62 J at a height (h) of 85 m. We need to find the mass (m) of the ball. Here's how we can solve it:

  1. Write down the known quantities:

    • GPE = 116.62 J
    • h = 85 m
    • g = 9.8 m/s² (acceleration due to gravity)

  2. Write down the formula for gravitational potential energy:

    • GPE = mgh

  3. Rearrange the formula to solve for mass (m):

    • To isolate m, we need to divide both sides of the equation by gh:
    • m = GPE / (gh)

The rearrangement of the formula is a crucial step in solving for the unknown variable. It demonstrates our ability to manipulate equations and apply mathematical principles to physical problems. The rearranged formula, m = GPE / (gh), allows us to directly calculate the mass by substituting the known values. The ability to rearrange formulas is a fundamental skill in physics and is essential for solving a wide range of problems. It requires a solid understanding of algebraic principles and the relationships between physical quantities. This step is not just about manipulating symbols; it's about understanding the underlying physics and how the variables interact. By rearranging the formula, we are essentially changing the perspective from which we view the relationship between GPE, mass, height, and gravitational acceleration. We are now focusing on mass as the unknown and expressing it in terms of the known quantities. This shift in perspective is a powerful problem-solving technique in physics. Furthermore, rearranging the formula helps us to identify the factors that directly influence the mass. We can see that the mass is directly proportional to the GPE and inversely proportional to the height and gravitational acceleration. This understanding provides insights into the physical meaning of the equation and how the variables are related. In summary, rearranging the formula is a critical step in solving for the mass. It demonstrates our ability to manipulate equations, apply mathematical principles, and gain a deeper understanding of the physical relationships between the variables. This skill is essential for solving a wide range of physics problems and for developing a strong foundation in the subject.

  1. Substitute the known values into the rearranged formula:
    • m = 116.62 J / (9.8 m/s² * 85 m)

The substitution of values is the stage where we bring together the theoretical framework and the specific details of the problem. It involves carefully replacing the symbolic variables in our rearranged formula with their corresponding numerical values, ensuring that the units are consistent. This step is not merely a mechanical process; it requires a clear understanding of what each variable represents and the units in which it is measured. When we substitute the values, we are essentially translating the abstract equation into a concrete calculation that is specific to our problem. The GPE of 116.62 J, the gravitational acceleration of 9.8 m/s², and the height of 85 m are now the building blocks of our solution. The accuracy of our final answer hinges on the precision of these substitutions. Any error in this step, such as using the wrong value or incorrect units, will propagate through the rest of the calculation and lead to an incorrect result. Therefore, meticulous attention to detail is paramount. Before substituting, it is always a good practice to double-check the values and their units to ensure that they are consistent with the formula and with each other. For instance, in this case, the GPE is in Joules, the gravitational acceleration is in meters per second squared, and the height is in meters, which are all consistent units for calculating mass in kilograms. The substitution process also provides an opportunity to check the reasonableness of the values. Are the given values within a realistic range for the scenario described? This kind of critical thinking can help identify potential errors or inconsistencies in the problem statement or in our understanding of the situation. In addition to numerical values, it is also important to include the units in the substitution process. This not only helps to ensure that the units are consistent but also provides a valuable check on the correctness of the formula and the calculation. By including the units, we can see how they combine and cancel out, leading to the final unit of the answer. In our case, the units will combine to give kilograms, which is the correct unit for mass. In summary, the substitution of values is a critical step in the problem-solving process. It requires careful attention to detail, a clear understanding of the variables and their units, and a critical assessment of the reasonableness of the values. This step transforms the abstract equation into a concrete calculation, bringing us closer to the solution.

  1. Calculate the mass:
    • m = 116.62 J / (833 m²/s²)
    • m ≈ 0.14 kg

The calculation of the mass is the culmination of our efforts, where we transform the numerical expression into a final, quantifiable answer. This step involves performing the arithmetic operations specified in the formula, paying close attention to the order of operations and the precision of the values. The result of this calculation is the mass of the ball, expressed in kilograms, which is the quantity we set out to determine. The arithmetic involved in this calculation is relatively straightforward, but it is crucial to perform it accurately. The numerator, 116.62 J, represents the gravitational potential energy of the ball, while the denominator, 833 m²/s², is the product of the gravitational acceleration and the height. Dividing the GPE by this product yields the mass. The use of a calculator is often necessary to ensure accuracy, especially when dealing with decimal values. However, it is equally important to understand the underlying arithmetic principles and to be able to estimate the answer mentally as a check against calculator errors. The order of operations is critical in this calculation. We must first multiply the gravitational acceleration and the height, and then divide the GPE by this product. Failing to follow the correct order of operations can lead to a significantly different and incorrect answer. The units also play a crucial role in this step. The units of GPE are Joules (J), which are equivalent to kg * m²/s². When we divide Joules by m²/s², the m²/s² units cancel out, leaving us with kilograms (kg), which is the correct unit for mass. This unit consistency provides a valuable check on the correctness of our calculation. The final answer, approximately 0.14 kg, represents the mass of the ball that possesses 116.62 J of gravitational potential energy at a height of 85 meters. This value is the tangible result of our problem-solving process, and it answers the question posed in the problem statement. However, it is not just a numerical value; it has physical meaning. It tells us something about the object we are studying and its relationship to the gravitational field. In summary, the calculation of the mass is the final step in our problem-solving journey. It involves performing the arithmetic operations with care, paying attention to the order of operations and the units, and interpreting the result in the context of the problem. This step transforms the numerical expression into a meaningful answer, providing us with a deeper understanding of the physical situation.

Answer

The mass of the ball is approximately 0.14 kg. Therefore, the correct answer is A. 0.14 kg.

Conclusion

In conclusion, we have successfully determined the mass of the ball by applying the formula for gravitational potential energy. By understanding the relationship between GPE, mass, height, and gravitational acceleration, we can solve a variety of physics problems. This problem highlights the importance of understanding fundamental physics concepts and applying them systematically. The ability to solve problems involving gravitational potential energy is a fundamental skill in physics and has applications in various fields, including engineering, mechanics, and astrophysics. The process we followed in this article, from understanding the concept of GPE to rearranging the formula and substituting values, is a general approach that can be applied to many physics problems. The key is to break down the problem into smaller, manageable steps, identify the known and unknown quantities, and apply the appropriate formulas and principles. Understanding the relationship between gravitational potential energy, mass, height, and gravitational acceleration is not just about solving problems; it's about gaining a deeper appreciation for the laws of nature that govern the world around us. The fact that we can predict the mass of an object based on its GPE and height is a testament to the power of physics as a predictive science. The ability to apply these principles in real-world scenarios is what makes physics so valuable and relevant. The problem we solved in this article is a simple example, but the principles involved are applicable to more complex situations. For instance, in designing roller coasters, engineers must carefully consider the GPE of the cars at the top of the hills and how it will be converted into kinetic energy as they descend. Similarly, in satellite mechanics, understanding the GPE of a satellite in orbit is crucial for determining its trajectory and ensuring its stability. In summary, understanding and applying the formula for gravitational potential energy is a fundamental skill in physics with wide-ranging applications. By mastering this concept and the problem-solving techniques we have discussed, you will be well-equipped to tackle a variety of physics challenges and to gain a deeper understanding of the physical world.