Chapter 11: Problem 1
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
Expert verified
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
01
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 (\(n\)): \(10\)
02
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: \(h\) = \(3 \mathrm{~m}\) × \(10\) = \(30 \mathrm{~m}\).
03
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:
\(PE = (70 \mathrm{~kg}) \times (9.81 \mathrm{m/s^2}) \times (30 \mathrm{~m})\)
04
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 \(3 \mathrm{~m}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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
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)
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.
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