Calculating Gravitational Potential Energy A 70 Kg Object At 8 Meters

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Let's delve into the concept of gravitational potential energy with a practical example. Imagine an object with a mass of 70 kilograms suspended at a height of 8 meters above the ground. This scenario provides a perfect opportunity to understand how gravitational potential energy works and how to calculate it. Gravitational potential energy, often abbreviated as GPE, is the energy an object possesses due to its position in a gravitational field. It’s the stored energy that has the potential to be converted into kinetic energy, which is the energy of motion. The higher an object is and the more massive it is, the more gravitational potential energy it has. This energy is relative to a reference point, which is often the ground but can be any arbitrary level.

When we talk about potential energy, we are really referring to the energy an object has because of its position relative to some other object. In this case, we are interested in the Earth's gravitational field and how it interacts with the 70 kg object. The Earth exerts a gravitational force on the object, and the object, in turn, has the potential to do work as it falls back to the ground. This potential to do work is what we quantify as gravitational potential energy. To fully grasp this concept, it’s essential to understand the factors that influence GPE: mass, height, and the acceleration due to gravity. Mass, measured in kilograms, is a fundamental property of matter that indicates its resistance to acceleration. Height, typically measured in meters, is the vertical distance of the object from the reference point. And the acceleration due to gravity, denoted as 'g', is the constant acceleration experienced by objects due to the Earth's gravitational pull, approximately 9.8 m/s². These three components are crucial in determining the gravitational potential energy of an object.

The significance of gravitational potential energy extends beyond theoretical physics; it has practical implications in various real-world applications. For instance, in hydroelectric power plants, water stored at a height possesses gravitational potential energy. As this water flows down, the GPE is converted into kinetic energy, which then drives turbines to generate electricity. Similarly, in roller coasters, the cars gain gravitational potential energy as they climb to the highest point, which is then converted into kinetic energy as they descend, providing the thrilling ride. Understanding GPE is also crucial in engineering, particularly in the design of structures and systems involving heights and masses, ensuring stability and safety. In essence, gravitational potential energy is a fundamental concept that underpins numerous phenomena and technologies we encounter daily.

To calculate the gravitational potential energy (GPE) of an object, we use a straightforward formula that incorporates the key factors we discussed: mass, height, and the acceleration due to gravity. The formula is:

GPE = m * g * h

Where:

  • GPE is 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 meters per second squared (m/s²).
  • h is the height of the object above the reference point, measured in meters (m).

This formula provides a clear and concise way to determine the potential energy stored in an object due to its position in a gravitational field. By plugging in the values for mass, gravity, and height, we can calculate the GPE in joules, which is the standard unit of energy in the International System of Units (SI). Understanding this formula is crucial for solving a wide range of physics problems and for appreciating how potential energy works in real-world scenarios. Now, let’s apply this formula to our specific example of the 70 kg object suspended at 8 meters.

Applying the GPE formula, we substitute the given values: mass (m) = 70 kg, height (h) = 8 meters, and the acceleration due to gravity (g) = 9.8 m/s². Plugging these values into the formula, we get:

GPE = 70 kg * 9.8 m/s² * 8 m

By performing this calculation, we can determine the gravitational potential energy of the object relative to the ground. This calculation is a direct application of the formula and provides a numerical answer that quantifies the stored energy. The result will be in joules, representing the amount of energy the object has due to its position in the Earth's gravitational field. Understanding how to perform this calculation is essential for anyone studying physics or dealing with practical applications involving potential energy.

Now, let's calculate the gravitational potential energy using the formula GPE = m * g * h. We have:

  • m = 70 kg
  • g = 9.8 m/s²
  • h = 8 m

Substituting these values into the formula:

GPE = 70 kg * 9.8 m/s² * 8 m
GPE = 5488 J

Therefore, the gravitational potential energy of the object with respect to the ground is 5,488 joules. This result means that the object has 5,488 joules of stored energy due to its position at 8 meters above the ground. This energy could be converted into kinetic energy if the object were allowed to fall, and it highlights the importance of understanding potential energy in various physical scenarios. The correct answer, therefore, is A) 5,488 J.

This calculation demonstrates a practical application of the gravitational potential energy formula and provides a clear understanding of how to quantify the energy stored in an object due to its height and mass. The result, 5,488 joules, is a specific value that represents the potential energy in this particular scenario. Understanding this calculation is crucial for solving similar problems and for appreciating the role of potential energy in different physical systems.

It’s important not only to find the correct answer but also to understand why the other options are incorrect. This can reinforce your understanding of the concepts and prevent similar mistakes in the future. In this case, the incorrect options are:

  • B) 560 J
  • C) 2,058 J
  • D) 3,430 J

Each of these options represents a miscalculation or a misunderstanding of the formula for gravitational potential energy. Let’s examine why each one is wrong.

Option B, 560 J, is likely the result of only multiplying the mass and the acceleration due to gravity, without considering the height. This calculation would be:

70 kg * 9.8 m/s² = 686 N (approximately)

This result is close to 560 J if rounded incorrectly or if some other calculation error was made. However, it misses the crucial step of multiplying by the height, which is essential for determining potential energy. This mistake highlights the importance of ensuring all components of the formula are included in the calculation.

Option C, 2,058 J, may arise from multiplying the mass by the height and using an incorrect value for the acceleration due to gravity. For example, if the acceleration due to gravity was mistakenly taken as 3 m/s², the calculation would be:

70 kg * 3 m/s² * 8 m = 1680 J

This result is not exactly 2,058 J, but it demonstrates how using an incorrect value for 'g' can lead to a wrong answer. It underscores the necessity of using the correct value for the acceleration due to gravity (9.8 m/s²) in the calculation.

Option D, 3,430 J, could be the result of a more complex miscalculation, such as dividing instead of multiplying, or using an incorrect combination of the given values. It is difficult to pinpoint the exact error without knowing the specific steps taken, but it is clear that the calculation does not correctly apply the gravitational potential energy formula. This reinforces the importance of carefully checking each step of the calculation to avoid errors.

In summary, understanding why these options are incorrect reinforces the correct method for calculating gravitational potential energy and highlights common mistakes to avoid. By recognizing these errors, you can strengthen your grasp of the concept and improve your problem-solving skills in physics.

In conclusion, the gravitational potential energy of the 70 kg object at a height of 8 meters above the ground is 5,488 J. This result is obtained by correctly applying the formula GPE = m * g * h, where mass, gravity, and height are accurately considered. Understanding this concept and the formula is essential for solving problems related to potential energy in physics.

Furthermore, recognizing why the other options are incorrect reinforces the importance of precise calculations and a thorough understanding of the formula. Avoiding common mistakes, such as omitting a component of the formula or using incorrect values, is crucial for achieving accurate results. The concept of gravitational potential energy has wide-ranging applications, from simple scenarios to complex engineering problems, making it a fundamental topic in physics. Mastering this concept not only helps in academic pursuits but also provides a valuable understanding of the physical world around us.