Use the table of 7.3 to find the point between the Earth and the Sun at which an object can be placed so that the net gravitational force exerted on it by these two objects is zero.
TABLE 7.3 <--- for the earth and the sun
Body
1.493e11
To find the point between the Earth and the Sun where the net gravitational force is zero, we need to locate the position where the gravitational forces from both objects cancel each other out.
In this case, we'll use the data from Table 7.3 to calculate the ratio of the gravitational forces between the Earth and the Sun at different distances.
The table you provided is missing, so I'll provide some example values for the distances from the Earth to the Sun and the corresponding gravitational forces:
Distance from the Earth to the Sun (in millions of kilometers) Gravitational Force (in Newtons)
100 36
200 9
300 4
400 2.25
500 1.44
We can see that as the distance increases, the gravitational force decreases.
To find the point where the net gravitational force is zero, we need to locate the distance where the forces from the Earth and the Sun balance each other. This occurs when the gravitational force from the Earth is equal in magnitude but opposite in direction to the gravitational force from the Sun.
Let's assume the distance at which the net gravitational force is zero is D.
At this distance, the gravitational force of the Earth and the gravitational force of the Sun will be equal in magnitude and opposite in direction:
|Force of the Earth| = |Force of the Sun|
Using the example values, we can try calculating the force ratio for different distances:
|36| = |Force of the Sun| (at 100 million kilometers)
|9| = |Force of the Sun| (at 200 million kilometers)
|4| = |Force of the Sun| (at 300 million kilometers)
It appears that the net gravitational force will not cancel out exactly at a specific distance in the given example values.
However, in reality, we know that the gravitational forces between astronomical objects follow an inverse square law. This means that the force decreases with the square of the distance.
If we had the full table, we could continue calculating the force ratio for different distances and locate the point where the net gravitational force would be zero.
To find the point between the Earth and the Sun at which the net gravitational force is zero, we can use the concept of gravitational balance.
In Table 7.3, we need information about the masses of the Earth and the Sun. If we assume the masses to be M (mass of the Sun) and m (mass of the Earth), we can use Newton's law of universal gravitation to find the point of balance.
According to Newton's law of universal gravitation, the gravitational force between two objects can be calculated using the equation:
F = G * (M * m) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2),
M is the mass of the Sun,
m is the mass of the Earth, and
r is the distance between the two objects.
To find the point of balance, we need to find the distance at which the gravitational forces from the Sun and the Earth cancel each other out (net force is zero).
1. Set up the equation for the gravitational forces from the Sun and the Earth:
F_s = G * (M * m) / (r_s)^2 (Force due to the Sun)
F_e = G * (M * m) / (r_e)^2 (Force due to the Earth)
2. To cancel out each other, the magnitudes of these forces should be equal:
F_s = F_e
3. Substitute the values of G, M, and m into the equation:
G * (M * m) / (r_s)^2 = G * (M * m) / (r_e)^2
4. Cross multiply the equation, cancel out the common terms:
(r_e)^2 = (r_s)^2
5. Simplify the equation and take the square root of both sides:
r_e = r_s
This means that the distance from the Earth to the Sun is equal to the distance from the Sun to the Earth, which is not a specific point, but the average distance between them, known as one Astronomical Unit (AU), which is approximately 149.6 million kilometers or 93 million miles.
Therefore, an object placed at a distance of one AU from the Sun (between the Earth and the Sun) experiences a net gravitational force of zero from these two objects.