Problem 1 - University Physics with Modern Physics

Question

In one day, a 75kg mountain climber ascends from the 1500m level on a vertical cliff to the top at 2400 m. The next day, she descends from the top to the base of the cliff, which is at an elevation of 1350 m. What is her change in gravitational potential energy (a) on the first day and (b) on the second day?

Step-by-Step Solution

Answer

First day: 662175 J; Second day: -772875 J.

Step 1: Understand Gravitational Potential Energy

Gravitational potential energy (GPE) is given by the formula \( PE = mgh \), where \( m \) is mass, \( g \) is acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \) on Earth), and \( h \) is height above a reference point. We need to calculate it for each specific change in altitude.

Step 2: Calculate GPE on the First Day

On the first day, the change in GPE is determined by the difference in the initial and final heights. Use \( h_1 = 1500 \, \text{m} \) and \( h_2 = 2400 \, \text{m} \) for calculations. Thus, \( \Delta h = 2400 - 1500 = 900 \, \text{m} \). The GPE change is \( \Delta PE = mg \Delta h = 75 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 900 \, \text{m} \).

Step 3: Compute the Energy Increase for the First Day

Substitute the values into the equation: \( \Delta PE = 75 \times 9.81 \times 900 = 662175 \, \text{J} \). Therefore, the change in gravitational potential energy on the first day is 662175 Joules.

Step 4: Calculate GPE on the Second Day

On the second day, the climber descends. The initial height \( h_1 \) is 2400 m, and the final height \( h_2 \) is 1350 m. Calculate the change in height: \( \Delta h = 1350 - 2400 = -1050 \, \text{m} \). The GPE change is \( \Delta PE = mg \Delta h = 75 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times (-1050) \, \text{m} \).

Step 5: Compute the Energy Decrease for the Second Day

Substitute the values into the equation: \( \Delta PE = 75 \times 9.81 \times (-1050) = -772875 \, \text{J} \). Therefore, the change in gravitational potential energy on the second day is -772875 Joules.

Key Concepts

Energy Calculation

Energy calculation is an important aspect in physics, especially when dealing with gravitational potential energy. To calculate gravitational potential energy (GPE), the formula used is \( PE = mgh \), where:

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

To find the change in GPE, we can look at the change in height \( \Delta h \). The formula becomes \( \Delta PE = mg \Delta h \). When calculating energy changes, remember:

• Positive \( \Delta h \) indicates an increase in height, resulting in an increase in GPE.
• Negative \( \Delta h \) indicates a decrease in height, corresponding to a decrease in GPE.

For example, in the case of our mountain climber, calculating energy changes on different days involves substituting these values into the formula for each segment of the journey.

Physics Problem Solving

Physics problem solving often involves breaking down a problem into manageable steps. By doing so, we can systematically approach and solve complex issues, such as calculating changes in gravitational potential energy. Here’s a simple strategy to navigate through such problems:

• **Understand the Concept**: Clearly identify what you are asked to find. In this case, it’s the change in gravitational potential energy.
• **Identify Known Variables**: Establish the necessary values, such as mass, height changes, and the acceleration due to gravity.
• **Apply the Right Formula**: Use the appropriate formula, like \( PE = mgh \), and adapt it based on what the problem requires.
• **Perform Calculations**: Substitute your known values into the formula and solve for the unknown.
• **Check Your Work**: Ensure that your solution makes sense in the real world. Consider units and the answer's logical soundness.

This structured approach not only leads to the right solution but also fosters a deeper understanding of underlying principles.

Mechanics

Mechanics is the branch of physics that deals with motion and the forces that affect motion. One of its subfields is the study of energy, including gravitational potential energy. Mechanics provides a framework to understand how physical systems behave, especially concerning energy transformations. Gravitational potential energy is a key concept in mechanics, embodying the energy an object possesses due to its position relative to the Earth. When an object is elevated to a higher altitude, it gains gravitational potential energy.

• Such energy calculations are crucial for assessing work and energy transfer in real-world scenarios, like our mountain climber problem.
• Mechanics helps us understand how energy is conserved and transformed from one form to another.
• It also illustrates the balance of forces acting on objects, which can result in a potential energy change.

Appreciating these principles can help students see the connections between theoretical calculations and actual physical events.