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8. Centripetal Forces & Gravitation

Gravitational Potential Energy for Systems of Masses

Physics

8. Centripetal Forces & Gravitation

Gravitational Potential Energy for Systems of Masses

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5:31 minutes

Problem 37

Textbook Question

What is the total gravitational potential energy of the three masses in FIGURE P13.35?

Verified step by step guidance

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.

Step 2: Identify the three masses and their positions from FIGURE P13.35. 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.

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.

Step 4: Add the gravitational potential energies of all pairs to find the total gravitational potential energy of the system. Use the equation:

U

_total = U₁₂ + U₁₃ + U₂₃.

Step 5: Ensure all values are substituted correctly, including the gravitational constant

G

, masses, and distances. Verify the units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the position of masses affects their potential energy in a gravitational system.

Superposition Principle

The superposition principle states that the total gravitational potential energy of a system of masses is the sum of the potential energies of each mass relative to a reference point. This means that when calculating the total gravitational potential energy for multiple masses, one must consider the individual contributions from each mass based on their respective heights and distances from the reference point.

Reference Point in Gravitational Calculations

In gravitational calculations, the choice of a reference point is essential as it determines the height (h) used in the potential energy formula. The reference point can be any location, but it is often chosen to be the ground or the lowest point in the system. The potential energy is relative, meaning that it can vary based on the selected reference point, affecting the total energy calculation for the system.

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