Question

(a) How far from the center of Earth would the net gravitational force of Earth and the Moon on an object be zero? (b) Setting the magnitudes of the forces equal should result in two answers from the quadratic. Do you understand why there are two positions, but only one where the net force is zero?

a) (a) About 3478 km, (b) Yes, due to gravitational forces acting in opposite directions.
b) (a) About 870 km, (b) Yes, due to the inverse square law of gravity.
c) Both of the above
d) None of the above

Answer 1

(a) About 3478 km, (b) Yes, due to gravitational forces acting in opposite directions.

Step-by-step explanation:

The point where the net gravitational force of Earth and the Moon on an object is zero can be determined by setting the magnitudes of their gravitational forces equal. This occurs when the gravitational pull from the Earth is balanced by the gravitational pull from the Moon, resulting in a net force of zero. The distance from the center of the Earth to this point is approximately 3478 km.

Gravitational force is given by Newton's law of gravitation, which states that the force is inversely proportional to the square of the distance between two masses. By equating the gravitational forces of the Earth and the Moon and solving for the distance, we find the point where these forces cancel each other out. The inverse square law of gravity contributes to the quadratic equation, leading to two solutions. However, only one of these positions corresponds to a situation where the net force is genuinely zero, as the forces are acting in opposite directions at this specific distance.

Answer 2

(a) The distance from the center of Earth where the net gravitational force of Earth and the Moon on an object is zero is about 3478 km. (b) Yes, there are two positions due to gravitational forces acting in opposite directions. Thus, the correct option is A.

Step-by-step explanation:

Firstly, to determine the distance where the net gravitational force is zero, we consider the gravitational forces exerted by the Earth and the Moon on an object. At this point, the gravitational force from the Earth and the gravitational force from the Moon are equal in magnitude but opposite in direction. Applying Newton's law of gravitation, the equation for the net force is
`\( F_{\text{net}} = \frac{{G \cdot M_{\text{Earth}} \cdot m}}{{(R - d)^2}} - \frac{{G \cdot M_{\text{Moon}} \cdot m}}{{d^2}} \), where \( G \) is the gravitational constant, \( M_{\text{Earth}} \) and \( M_{\text{Moon}} \)`are the masses of Earth and the Moon,m is the mass of the object,R is the distance from the center of Earth to the object, and d is the distance between the object and the center of the Moon. Solving for d, we find
`\( d \approx 3478 \) km.`

Secondly, understanding why there are two positions involves recognizing that there are two points along the line connecting the centers of the Earth and the Moon where the gravitational forces balance. These points are where the gravitational forces from the Earth and the Moon act in opposite directions, resulting in a net force of zero. The inverse square law of gravity implies that the gravitational force weakens as the distance increases, leading to two points where the forces balance but only one where the net force is zero.

In conclusion, the correct answer is (a) About 3478 km; (b) Yes, due to gravitational forces acting in opposite directions. The explanation involves solving for the distance where the net gravitational force is zero and understanding that two positions satisfy this condition along the line connecting the Earth and the Moon.