Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 37

Chapter 13, Problem 37

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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Two spherical objects have a combined mass of 150 kg. The gravitational attraction between them is 8.00 x 10^-6 N when their centers are 20 cm apart. What is the mass of each?

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Textbook Question

What is the net gravitational force on the 20.0 kg mass in FIGURE P13.36? Give your answer using unit vectors.

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Two spherical objects have a combined mass of 400 kg. The gravitational attraction between them is 6.00 x 10^-7 N and their gravitational potential energy is ₋1.20 x 10^-6 J. What is the mass of each?

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The Parker Solar Probe, launched in 2018, was the first spacecraft to explore the solar corona, the hot gases and flares that extend outward from the solar surface. The probe is in a highly elliptical orbit that, using the gravity of Venus, will be nudged ever closer to the sun until, in 2025, it reaches a closest approach of 6.9 million kilometers from the center of the sun. Its maximum speed as it whips through the corona will be 192 km/s. The probe's highly elliptical orbit carries it out to a maximum distance of 160 R_s with a period of 88 days. What is its slowest speed, in km/s?

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Textbook Question

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