Potential Energy Calculation For A Man Climbing Stairs With A Package

Jun 28, 2025 by ADMIN 70 views

A man with a mass of 80kg carrying a 30kg package climbs a staircase with 20 steps, each 15 cm high. What is the potential energy of the system when the man reaches the top of the staircase? Consider g = 10m/s².

Calculating Potential Energy A Man Climbs a Staircase with a Load

In the realm of physics, understanding potential energy is crucial for comprehending how energy is stored within a system due to its position or configuration. This article delves into a practical scenario involving a man climbing a staircase while carrying a load, aiming to calculate the total potential energy of the system at the highest point. We will explore the concepts of gravitational potential energy, the factors influencing it, and the step-by-step calculations required to arrive at the solution. By analyzing this example, we can gain valuable insights into the application of potential energy principles in real-world situations.

Consider a man with a mass of 80 kg carrying a 30 kg package. He ascends a staircase consisting of 20 steps, each 15 cm high. Our objective is to determine the gravitational potential energy of the system (man and package) when the man reaches the top of the staircase, assuming the acceleration due to gravity (g) is 10 m/s². This problem allows us to apply the fundamental principles of potential energy and understand how it changes with height and mass. By solving this, we will understand the relationship between mass, gravity, height, and potential energy.

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It's the stored energy that can be converted into other forms of energy, such as kinetic energy, when the object is allowed to move freely under the influence of gravity. The higher an object is and the more massive it is, the more gravitational potential energy it possesses. Understanding this concept is vital in various fields, including engineering, physics, and even everyday life, as it helps us analyze and predict the behavior of objects in a gravitational field.

The formula for gravitational potential energy is given by:

GPE = m * g * h

Where:

GPE is the gravitational potential energy (measured in Joules)
m is the mass of the object (measured in kilograms)
g is the acceleration due to gravity (approximately 9.8 m/s² on Earth, but we'll use 10 m/s² as given in the problem)
h is the vertical height above a reference point (measured in meters)

This formula highlights the direct relationship between gravitational potential energy, mass, and height. A larger mass or a greater height will result in a higher gravitational potential energy. The value of 'g', the acceleration due to gravity, remains relatively constant near the Earth's surface, making mass and height the primary variables influencing GPE in most practical scenarios.

Calculating Total Mass

The first step in calculating the potential energy of the system is to determine the total mass being lifted against gravity. This includes both the mass of the man and the mass of the package he is carrying. By combining these masses, we obtain the total mass, which is a crucial component in the potential energy calculation. This step ensures that we account for all the mass contributing to the system's potential energy at the top of the staircase.

Mass of the man (m₁) = 80 kg
Mass of the package (m₂) = 30 kg
Total mass (m) = m₁ + m₂ = 80 kg + 30 kg = 110 kg

Therefore, the total mass of the system, which includes the man and the package, is 110 kg. This total mass will be used in the potential energy formula to determine the energy stored due to the system's position at the top of the staircase. It's essential to consider the total mass to accurately assess the gravitational potential energy.

Determining the Total Height

To calculate the potential energy, we need to find the total vertical height the man ascends. This height is determined by the number of steps and the height of each step. By multiplying these two values, we can find the total height, which is a critical parameter in the potential energy calculation. Ensuring accuracy in this step is crucial, as the height directly impacts the calculated potential energy value.

Number of steps = 20
Height of each step = 15 cm = 0.15 meters (converting cm to meters)
Total height (h) = Number of steps × Height of each step
h = 20 × 0.15 meters = 3 meters

Thus, the total vertical height the man ascends is 3 meters. This height represents the vertical displacement against gravity and will be used in conjunction with the total mass and the acceleration due to gravity to calculate the gravitational potential energy of the system at the top of the staircase. The conversion from centimeters to meters is essential to maintain consistent units in the calculation.

Now that we have the total mass of the system (110 kg), the total height ascended (3 meters), and the acceleration due to gravity (10 m/s²), we can calculate the gravitational potential energy using the formula:

GPE = m * g * h

Substituting the values we have:

GPE = 110 kg * 10 m/s² * 3 meters
GPE = 3300 Joules

Therefore, the gravitational potential energy of the system when the man reaches the top of the staircase is 3300 Joules. This value represents the energy stored in the system due to its position in the Earth's gravitational field. The calculation underscores the relationship between mass, height, gravity, and potential energy, providing a clear understanding of how energy is stored in a system.

In summary, the gravitational potential energy of the system, consisting of an 80 kg man carrying a 30 kg package up a 20-step staircase (each step being 15 cm high), is 3300 Joules. This calculation highlights the practical application of the potential energy formula and demonstrates how it is used to determine the energy stored in a system due to its position within a gravitational field. By breaking down the problem into steps—calculating total mass, determining total height, and then applying the formula—we arrive at a clear and accurate solution.

Understanding gravitational potential energy is essential in various fields, from physics and engineering to everyday life. This example illustrates how potential energy is influenced by mass, height, and gravity, providing a fundamental understanding of energy storage and transformation. By mastering these concepts, we can better analyze and predict the behavior of objects in various physical systems. The ability to calculate potential energy accurately is not only a valuable skill in scientific and engineering contexts but also enhances our understanding of the world around us.