Given that the distance between the Earth and the Moon is d = 3.84 108 m, show
that a satellite located exactly in-between the Earth and the Moon at a distance of
90% d from the Earth experiences no net force (at least when only the
gravitational force due to the Earth and the Moon at taken into account). Draw a
diagram showing the forces acting on the satellite.
repeat.
To show that a satellite located exactly halfway between the Earth and the Moon at a distance of 90% d from the Earth experiences no net force, we need to consider the gravitational forces acting on it from both the Earth and the Moon.
The force of gravity between two objects is given by the gravitational force equation:
F = G * (m1 * m2) / r^2
Where:
- F is the force of gravity
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
Let's denote the mass of the Earth as M1, the mass of the Moon as M2, and the distance between the Earth and the Moon as d.
1. Calculate the force of gravity exerted by the Earth on the satellite.
The distance between the Earth and the satellite is 0.9d, and the mass of the satellite is negligible compared to the Earth. Therefore, we can assume the mass of the satellite as zero.
Force of gravity exerted by the Earth on the satellite:
F1 = G * (M1 * 0) / (0.9d)^2
F1 = 0 N
Since the mass of the satellite is negligible, the gravitational force exerted by the Earth on the satellite cancels out.
2. Calculate the force of gravity exerted by the Moon on the satellite.
The distance between the Moon and the satellite is 0.1d. As before, we can assume the mass of the satellite as zero.
Force of gravity exerted by the Moon on the satellite:
F2 = G * (M2 * 0) / (0.1d)^2
F2 = 0 N
Similarly, the gravitational force exerted by the Moon on the satellite also cancels out due to the negligible mass of the satellite.
Since both forces cancel each other out, the satellite located exactly halfway between the Earth and the Moon at a distance of 90% d from the Earth experiences no net force.
To visualize the forces acting on the satellite, you can draw a diagram with the Earth, the Moon, and the satellite located between them. Label the forces acting on the satellite as F1 (force from Earth) and F2 (force from Moon), both of which are equal in magnitude but opposite in direction. This diagram will show that the forces balance each other out, resulting in no net force on the satellite.