As a moon follows its orbit around a planet, the maximum gravitational force exerted on the moon by the planet exceeds the minimum gravitational force by 9.8%. Find the ratio rmax/rmin, where rmax is the moon's maximum distance from the center of the planet and rmin is the minimum distance.
To find the ratio rmax/rmin, we need to determine the relationship between the maximum and minimum gravitational forces and use that to find the relationship between the maximum and minimum distances.
Let's assume that the maximum gravitational force is Fmax and the minimum gravitational force is Fmin. We are given that Fmax exceeds Fmin by 9.8%. Mathematically, we can write this as:
Fmax = Fmin + 0.098 * Fmin
Simplifying this equation, we have:
Fmax = 1.098 * Fmin
Now, let's take a look at the relationship between gravitational force and distance. According to Newton's law of universal gravitation, the gravitational force between two objects is inversely proportional to the square of the distance between them. Mathematically, we can write this as:
F = G * (m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.
Since the masses of the moon and the planet remain constant, we can simplify the equation to:
F ∝ 1 / r^2
Now, let's analyze the relationship between distances. Since the gravitational force increases as the distance decreases, we can conclude that the maximum distance corresponds to the minimum gravitational force, and the minimum distance corresponds to the maximum gravitational force. Mathematically, we can write this relationship as:
rmax ∝ 1 / Fmin
Similarly,
rmin ∝ 1 / Fmax
Applying the previously derived relationship between Fmax and Fmin, we have:
rmin ∝ 1 / (1.098 * Fmin)
rmax ∝ 1 / Fmin
To find the ratio rmax/rmin, we can divide the equation for rmax by the equation for rmin:
(rmax / rmin) = (1 / Fmin) / (1 / (1.098 * Fmin))
Simplifying this equation, we have:
(rmax / rmin) = (1.098 * Fmin) / Fmin
(rmax / rmin) = 1.098
Therefore, the ratio rmax/rmin is 1.098.