Skip to main content
Ch 13: Newton's Theory of Gravity
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 13: Newton's Theory of GravityProblema 6
Capitolo 13, Problema 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?

Guida verificata passo dopo passo
1
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 \( d \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
Percorso guidato
05:41
Gravitational Forces in 2D

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.
Video consigliato:
Percorso guidato
07:32
Weight Force & Gravitational Acceleration

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.
Video consigliato:
Percorso guidato
9:26
Gravitational Force Inside the Earth
Pratica correlata
Domanda del libro di testo

What is the ratio of the sun's gravitational force on you to the earth's gravitational force on you?

2043
views
Domanda del libro di testo

The International Space Station orbits 300 km above the surface of the earth. What is the gravitational force on a 1.0 kg sphere inside the International Space Station?

1641
views
1
rank
Domanda del libro di testo

A recently discovered extrasolar planet appears to be rockier and denser than earth. It is 16 times as massive as earth, but its diameter is only twice that of earth. What is the free-fall acceleration on the surface of this planet?

2276
views
Domanda del libro di testo

Asteroid 253 Mathilde is one of several that have been visited by space probes. This asteroid is roughly spherical with a diameter of 53 km. The free-fall acceleration at the surface is 9.9 ✕ 10-3 m/s2. What is the asteroid's mass?

1423
views
Domanda del libro di testo

What is the free-fall acceleration at the surface of (a) the moon and (b) Jupiter?

1869
views
1
rank
Domanda del libro di testo

What is the force of attraction between a 50 kg woman and a 70 kg man sitting 1.0 m apart?

1510
views