Question

If a 70 kg person climbs 10 flights of stairs, each flight \(3 \mathrm{~m}\) high, how much potential energy have they gained?

Short answer

Answer: The person has gained 20,574 Joules of potential energy after climbing 10 flights of stairs, each with a height of 3 meters.

Step-by-step solution

Identify the given information

From the problem, we are given the following information: - Mass of the person (\(m\)): \(70 \mathrm{~kg}\) - Height of each flight of stairs (\(h_1\)): \(3 \mathrm{~m}\) - Number of flights of stairs climbed (\(n\)): \(10\)

Calculate the total height climbed

To calculate the total height climbed, we need to multiply the height of each flight of stairs by the number of flights climbed: Total height (\(h\)) = height of each flight (\(h_1\)) × number of flights (\(n\)) In our case: \(h\) = \(3 \mathrm{~m}\) × \(10\) = \(30 \mathrm{~m}\).

Calculate the potential energy gained

Now, we can calculate the potential energy using the formula: \(PE = mgh\). We will use the following values: - Mass of the person (\(m\)): \(70 \mathrm{~kg}\) - Gravitational acceleration (\(g\)): \(9.81 \mathrm{m/s^2}\) - Total height climbed (\(h\)): \(30 \mathrm{~m}\) Plugging in the values, we get: \(PE = (70 \mathrm{~kg}) \times (9.81 \mathrm{m/s^2}) \times (30 \mathrm{~m})\)

Solve for potential energy

Finally, we will solve for the potential energy: \(PE = 70 \times 9.81 \times 30\) \(PE = 20574 \mathrm{~J}\) (Joules) The person has gained \(20,574\) Joules of potential energy after climbing \(10\) flights of stairs, each with a height of \(3 \mathrm{~m}\).

Key concepts

Gravitational Potential Energy

When you lift an object against the force of gravity, you store energy in that object. This stored energy is known as Gravitational Potential Energy (GPE). It is a type of potential energy that depends on the height of an object above the ground. The higher you lift the object, the more potential energy it gains. This energy can be calculated using the formula: \[ PE = mgh \] Where:

• \(m\) is the mass of the object in kilograms (kg)
• \(g\) is the acceleration due to gravity, approximately \(9.81 \text{ m/s}^2\) on Earth
• \(h\) is the height in meters (m) above the reference point

In our example, a person climbing stairs gains potential energy as they rise higher. The energy is proportional to the mass of the person, the height climbed, and the gravitational acceleration.

Physics Calculations

Physics calculations involve a series of logical steps to find an answer to a problem. Understanding the process behind these calculations is essential: To calculate Gravitational Potential Energy in the exercise, the first step is to gather all the relevant data from the problem, such as mass, gravitational acceleration, and height. Next, it's crucial to use the correct formulas and consistently apply units throughout the calculation.

Step-by-Step Calculation Process

• **Identify given information**: Mass \(m = 70\, kg\) and total height \(h = 30\, m\)
• **Use the formula**: \[ PE = mgh \]
• **Apply values**: Insert \(m = 70\, kg\), \(g = 9.81\, m/s^2\), and \(h = 30\, m\) into \[ PE = 70 \times 9.81 \times 30 \]
• **Calculate**: Perform the multiplication to find \(PE = 20574\, J\) (Joules)

The step-by-step approach helps ensure accuracy and can be applied to various physics problems.

Energy Conservation

Energy conservation is a fundamental principle in physics, stating that energy cannot be created or destroyed, only transformed from one form to another. When a person climbs stairs, chemical energy from food is converted into mechanical energy, which is then converted into Gravitational Potential Energy as they elevate.
This principle is key for understanding how energy flows and converts during different activities. For instance, when the person descends, the potential energy can transform into kinetic energy as they speed downward, continuing to obey the law of conservation.

Applications of Energy Conservation

• Used in understanding mechanisms of various physical phenomena
• Helps in solving complex problems by accounting for all energy types
• Integral for the design of machines and systems that efficiently manage energy use

Thus, acknowledging energy conservation provides a comprehensive view of energy transformations in all processes including everyday actions like climbing stairs.