What is the total gravitational potential energy of the three masses in FIGURE P13.36?

Verified step by step guidance

1. Step 1: Understand the concept of gravitational potential energy. Gravitational potential energy between two masses is given by the formula: $U=-\frac{G}{m}$, where $G$ is the gravitational constant, $m$ and $m$ are the masses, and $r$ is the distance between them.
2. Step 2: Identify the three masses and their positions from the figure. Determine the distances between each pair of masses. Label the masses as $m$₁, $m$₂, and $m$₃, and calculate the distances $r$₁₂, $r$₁₃, and $r$₂₃.
3. Step 3: Calculate the gravitational potential energy for each pair of masses using the formula $U=-\frac{G}{m}$. For example, calculate $U$₁₂, $U$₁₃, and $U$₂₃.
4. Step 4: Add the gravitational potential energies of all pairs to find the total gravitational potential energy. The total gravitational potential energy is given by: $U$_total=U₁₂+U₁₃+U₂₃.
5. Step 5: Ensure all values are substituted correctly, including the gravitational constant $G$, masses, and distances. Perform the addition to find the total gravitational potential energy.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (U) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height above a reference point. This energy is crucial for understanding how objects interact under the influence of gravity.

Mass and Weight

Mass is a measure of the amount of matter in an object, typically measured in kilograms. Weight, on the other hand, is the force exerted by gravity on that mass, calculated as W = mg. Understanding the distinction between mass and weight is essential for calculating gravitational potential energy, as the mass directly influences the energy value.

Reference Point in Potential Energy Calculations

In gravitational potential energy calculations, the choice of reference point is critical. The height (h) in the potential energy formula is measured relative to this reference point, which can be the ground, the lowest point of the system, or any other convenient level. Different reference points can yield different potential energy values, but the energy differences remain consistent.