Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 6

Chapter 13, Problem 6

Two 65 kg astronauts leave earth in a spacecraft, sitting 1.0 m apart. How far are they from the center of the earth when the gravitational force between them is as strong as the gravitational force of the earth on one of the astronauts?

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Identify the forces at play: The problem involves two gravitational forces. The first is the gravitational force between the two astronauts, and the second is the gravitational force exerted by the Earth on one astronaut. These forces must be equal in magnitude for the given condition.

Write the formula for the gravitational force between the two astronauts: \( F_{astronauts} = \frac{G \cdot m_1 \cdot m_2}{r^2} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the astronauts (both 65 kg), and \( r \) is the distance between them (1.0 m).

Write the formula for the gravitational force exerted by the Earth on one astronaut: \( F_{earth} = \frac{G \cdot M_{earth} \cdot m_{astronaut}}{d^2} \), where \( M_{earth} \) is the mass of the Earth, \( m_{astronaut} \) is the mass of one astronaut (65 kg), and \( d \) is the distance from the center of the Earth to the astronaut.

Set the two forces equal to each other: \( \frac{G \cdot m_1 \cdot m_2}{r^2} = \frac{G \cdot M_{earth} \cdot m_{astronaut}}{d^2} \). Simplify the equation by canceling \( G \) and substituting the known values for \( m_1 \), \( m_2 \), \( m_{astronaut} \), and \( r \).

Solve for \( d \): Rearrange the equation to isolate \( d \), resulting in \( d = \sqrt{\frac{M_{earth} \cdot r^2}{m_1}} \). Substitute the known values for \( M_{earth} \) (mass of the Earth), \( r \) (1.0 m), and \( m_1 \) (65 kg) to calculate \( d \).

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Key Concepts

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

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. This concept is crucial for understanding how the astronauts interact with each other and with the Earth.

Weight and Gravitational Acceleration

Weight is the force exerted on an object due to gravity, calculated as the product of mass and gravitational acceleration (W = mg). On Earth, the average gravitational acceleration is approximately 9.81 m/s². This concept helps in determining the gravitational force acting on the astronauts due to the Earth, which is essential for comparing it to the gravitational force between the astronauts.

Distance from the Center of the Earth

The distance from the center of the Earth is critical in calculating gravitational forces, as gravitational strength decreases with increasing distance. The formula for gravitational force includes the distance squared in the denominator, meaning that even small changes in distance can significantly affect the force. Understanding this concept is necessary to determine how far the astronauts must be from the Earth's center for the forces to balance.

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