Question

Analyze Is the change in potential energy when a box is lifted 1 meter off the ground the same on Earth and the Moon? Explain.

Short answer

No, the change in potential energy is greater on Earth than on the Moon due to higher gravity.

Step-by-step solution

Understand the Problem

We need to compare the change in gravitational potential energy when a box is lifted by 1 meter on both Earth and the Moon.

Recall the Formula for Gravitational Potential Energy

The formula for gravitational potential energy is \( U = mgh \), where \( U \) is potential energy, \( m \) is mass, \( g \) is the gravitational acceleration, and \( h \) is the height.

Identify Values for Gravitational Acceleration

On Earth, the gravitational acceleration \( g \) is approximately \( 9.8 \, \text{m/s}^2 \), while on the Moon, it is about \( 1.6 \, \text{m/s}^2 \).

Calculate Potential Energy Change on Earth

Let's calculate the change in potential energy, \( \Delta U \), for lifting a box by 1 meter on Earth. Assuming the mass \( m \) is constant:\[ \Delta U_{\text{Earth}} = mg_{\text{Earth}} \times 1 \approx m \times 9.8 \, \text{J} \]

Calculate Potential Energy Change on the Moon

We do the same calculation for the Moon:\[ \Delta U_{\text{Moon}} = mg_{\text{Moon}} \times 1 \approx m \times 1.6 \, \text{J} \]

Compare the Results

Compare the change in potential energy on Earth and the Moon:- \( \Delta U_{\text{Earth}} = m \times 9.8 \, \text{J} \)- \( \Delta U_{\text{Moon}} = m \times 1.6 \, \text{J} \)Since \( 9.8 > 1.6 \), the change in potential energy on Earth is greater than on the Moon when lifting the same box by 1 meter.

Key concepts

Gravitational Acceleration

Gravitational acceleration is the rate at which an object's velocity changes due to the force of gravity. It is a crucial component in calculations involving gravitational potential energy. On Earth, the average gravitational acceleration is approximately \( 9.8 \, \text{m/s}^2 \). This value is constant at all points near the surface of the Earth. However, on the Moon, this value is much smaller, about \( 1.6 \, \text{m/s}^2 \). This lower gravitational force is due to the Moon's smaller mass and size compared to Earth.

Understanding gravitational acceleration helps us predict how much force is necessary to move an object or how fast it accelerates when dropped. When comparing different celestial bodies, knowing their gravitational acceleration allows us to compare how strong their gravitational pull is. This is significant in potential energy calculations because gravitational acceleration directly affects how much potential energy an object gains when lifted.

When solving problems in physics, always check the gravitational acceleration values. They may vary depending on the planetary body you are dealing with, and this can dramatically alter your calculations.

Potential Energy Comparison

Potential energy refers to the energy an object has due to its position or state. The gravitational potential energy depends on three main factors: mass (\( m \)), gravitational acceleration (\( g \)), and height (\( h \)). The formula to calculate gravitational potential energy is \( U = mgh \).

When comparing potential energy changes, it is essential to recognize how different variables affect the outcome. For the exercise, lifting a box one meter on Earth gives a different potential energy change compared to doing the same on the Moon. This difference arises because the gravitational acceleration on Earth (\( 9.8 \, \text{m/s}^2 \)) is higher than on the Moon (\( 1.6 \, \text{m/s}^2 \)).

Here are some key points:

• The greater gravitational force on Earth means more potential energy is gained when lifting an object than on the Moon.
• Even with identical mass and lifting height, the potential energy change will vary based on the planet.
• This concept helps us understand why activities and movements differ between Earth and other planetary bodies.

Knowing these comparisons is useful not just in tackling homework problems but also in diverse fields like physics research and space exploration.

Physics Problem Solving

When facing physics problems, whether in class exercises or real-world applications, a structured problem-solving strategy can enhance understanding and lead to accurate outcomes. Breaking down a problem involves several steps:

1. **Understand the Problem**: Clearly explain what is being asked. In our exercise, the aim was to compare potential energy changes on Earth and the Moon.

2. **Recall Relevant Formulas**: Identify and write down the formulae related to the problem. For gravitational potential energy, knowledge of the formula \( U = mgh \) was central.

3. **Identify Given Values**: Determine the given values or those that can be deduced. In this case, gravitational acceleration values on Earth and the Moon were needed.

4. **Perform Calculations**: Systematically calculate step-by-step, being vigilant of unit consistency. Separate calculations for Earth's and the Moon's energy change allow direct comparison.

5. **Analyze and Compare Results**: Interpret your results to ensure they make sense. Check whether the potential energy difference aligns with the expected gravitational characteristics of the Earth and Moon.

This structured approach aids in dissecting complex problems into manageable parts, leading to a thorough understanding and coherent solutions. By consistently applying these steps, students can develop proficiency in tackling various physics challenges.