Chapter 04, Problem 032 Chalkboard Video A spacecraft is on a journey to the moon. At what point, as measured from the center of the earth, does the gravitational force exerted on the spacecraft by the earth balance that exerted by the moon? This point lies on a line between the centers of the earth and the moon. The distance between the earth and the moon is 3.85 × 108 m, and the mass of the earth is 81.4 times as great as that of the moon. moon earth
Gravitational force
In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.
Acceleration Due to Gravity
In fundamental physics, gravity or gravitational force is the universal attractive force acting between all the matters that exist or exhibit. It is the weakest known force. Therefore no internal changes in an object occurs due to this force. On the other hand, it has control over the trajectories of bodies in the solar system and in the universe due to its vast scope and universal action. The free fall of objects on Earth and the motions of celestial bodies, according to Newton, are both determined by the same force. It was Newton who put forward that the moon is held by a strong attractive force exerted by the Earth which makes it revolve in a straight line. He was sure that this force is similar to the downward force which Earth exerts on all the objects on it.
![### Gravitational Equilibrium Point between Earth and Moon
#### Problem Statement:
A spacecraft is on a journey to the moon. At what point, as measured from the center of the earth, does the gravitational force exerted on the spacecraft by the earth balance that exerted by the moon? This point lies on a line between the centers of the earth and the moon. The distance between the earth and the moon is \( 3.85 \times 10^8 \) meters, and the mass of the earth is 81.4 times as great as that of the moon.
#### Data Provided:
- Distance between Earth and Moon: \( 3.85 \times 10^8 \) meters
- Mass ratio of Earth to Moon: 81.4
#### Graphical Representation:
The image includes a schematic diagram depicting the Earth, Moon, and the point where the gravitational forces balance. It shows:
1. **Earth** (large blue and green circle).
2. **Moon** (smaller grey circle).
3. **Forces** acting on the spacecraft:
- \( F_{\text{earth}} \) directed towards the Earth.
- \( F_{\text{moon}} \) directed towards the Moon.
4. **Distance** between Earth and Moon denoted as \( 3.85 \times 10^8 \) meters.
5. **Distance** from the spacecraft to the Earth denoted as \( r \).
6. **Distance** from the spacecraft to the Moon denoted as \( 3.85 \times 10^8 - r \).
To find the exact point where the spacecraft's gravitational forces exerted by the Earth and the Moon are in equilibrium, one would solve the equation derived from Newton's law of gravitation:
\[ \frac{F_{\text{earth}}}{F_{\text{moon}}} = \frac{G \cdot M_{\text{earth}} / r^2}{G \cdot M_{\text{moon}} / (3.85 \times 10^8 - r)^2} = 1 \]
Given the mass ratio (81.4 times), we set up and solve the resulting equation:
\[ \frac{81.4}{1} = \left(\frac{3.85 \times 10^8 - r}{r}\right)^2 \]
By solving this equation, one can find the value of \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc783430-bf94-4e65-bed2-17698c45aa9d%2Fb665327c-a70c-4317-ae67-2ba7d2029911%2Fut95j9_processed.jpeg&w=3840&q=75)
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