Problem 40 - College Physics

Question

The floor of the basement of a house is \(3.0 \mathrm{~m}\) below ground level, and the floor of the attic is \(4.5 \mathrm{~m}\) above ground level. (a) If an object in the attic were brought to the basement, the change in potential energy will be greatest relative to which floor: (1) attic, (2) ground, (3) basement, or (4) all the same? Why? (b) What are the respective potential energies of \(1.5-\mathrm{kg}\) objects in the basement and attic, relative to ground level? (c) What is the change in potential energy if the object in the attic is brought to the basement?

Step-by-Step Solution

Answer

(a) Basement; (b) Basement: -44.145 J, Attic: 66.2175 J; (c) Change: -110.3625 J.

Step 1: Understanding the Problem

First, determine what the problem is asking. You need to find the greatest change in potential energy and calculate the potential energy at different locations relative to the ground.

Step 2: Analyzing Part (a)

For part (a), consider the potential energy relative to three locations: the attic, ground, and basement. The change in potential energy is greatest when the reference point is at the lowest potential position, rather than the higher ones.

Step 3: Solving Part (a)

Potential energy change is calculated as the difference in height multiplied by the gravitational force. The greatest change occurs when comparing the highest point (the attic) to the lowest point (the basement, which is the lowest possible potential position). Therefore, the change is greatest relative to the basement.

Step 4: Setting up for Part (b)

To find potential energies of the object when located at the basement and attic (relative to the ground), use the formula for gravitational potential energy: \[U = mgh\]where \(m\) is the mass, \(g\) is the gravitational acceleration \(9.81\, \text{m/s}^2\), and \(h\) is the height relative to the ground.

Step 5: Calculating Basement Potential Energy

For the basement, the height \(h = -3.0\) meters.\(U_{\text{basement}} = 1.5 \times 9.81 \times (-3.0)\)\(U_{\text{basement}} = -44.145\, \text{Joules}\).

Step 6: Calculating Attic Potential Energy

For the attic, the height \(h = 4.5\) meters.\(U_{\text{attic}} = 1.5 \times 9.81 \times 4.5\)\(U_{\text{attic}} = 66.2175\, \text{Joules}\).

Step 7: Solving Part (c)

The change in potential energy if the object moves from the attic to the basement is the difference between the potential energies calculated.\( \Delta U = U_{\text{basement}} - U_{\text{attic}}\)\( \Delta U = -44.145 - 66.2175\)\( \Delta U = -110.3625\, \text{Joules}\).

Step 8: Final Step: Conclusion

The motion from attic to basement results in the highest potential energy change relative to the basement. The potential energy at the basement is \(-44.145\, \text{Joules}\), at the attic is \(66.2175\, \text{Joules}\), and the change when moved is \(-110.3625\, \text{Joules}\).

Key Concepts

Understanding Gravitational Potential Energy

Gravitational potential energy is an essential concept in physics that helps us understand how energy is stored due to an object's position within a gravitational field. When an object is elevated from a reference point, it gains potential energy. This energy is calculated using the formula: \[U = mgh\]where:

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

In our example, the height is measured from a reference point, usually the ground, and involves calculating how far the object is from that point in the vertical direction. The greater the height (\(h\)), the more potential energy the object possesses.

Mastering Energy Calculations

Energy calculations involve determining the potential energy at various heights and understanding how this energy changes when the object's position shifts. For instance, the potential energy for an object at a reference point (like the ground) would be zero because its height, \(h\), is zero.For our problem, when the object is at the attic, 4.5 meters above the ground, the potential energy will be: \[U_{\text{attic}} = mgh = 1.5 \times 9.81 \times 4.5 = 66.2175\, \text{Joules}\]And when it's in the basement, 3.0 meters below the ground level, the potential energy is: \[U_{\text{basement}} = mgh = 1.5 \times 9.81 \times (-3.0) = -44.145\, \text{Joules}\]Calculating these values helps us understand the stored energy changes as the object moves between different heights. The change in energy is reflected in the difference in potential energy between two positions.

Techniques for Physics Problem-Solving

Successfully solving physics problems requires a clear approach to break down what is needed. Start by understanding what the problem asks. Here, it’s about determining potential energy differences based on location.

• Identify known quantities (mass, gravity, height differences).
• Calculate the potential energy for each location using the formula \(U = mgh\).
• Determine the change in potential energy by finding the difference between two specific energies.

In part (c) of the exercise, the change in energy when the object is moved from the attic to the basement is:\[ \Delta U = U_{\text{basement}} - U_{\text{attic}} = -44.145 - 66.2175 = -110.3625\, \text{Joules} \]This process illustrates systematic physics problem-solving, using given data and tried methods for accurate solutions.

Importance of Reference Points in Energy Calculations

Reference points are crucial in energy calculations because the potential energy depends on the position relative to a chosen baseline. These points let us specify areas to measure changes accurately as an object moves. In this scenario, using the ground level as a reference point is standard, providing clear measurements above or below it. However, the problem also suggests different reference points—basement or attic. When using the basement as reference, moving an object from attic to basement shows maximum energy change because the basement represents the lowest potential. Always choose reference points wisely:

• Ensure consistency in calculations,
• Avoid confusion with negative or positive heights,
• Clearly define your baseline to compare different energy states effectively.