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.
(rmin/rmax)^2 = Fmax/Fmin = 1.098
rmin/rmax = 1.0479
rmax/rmin = 0.9543
Thank You
To find the ratio rmax/rmin, we can first express the relationship between the maximum and minimum gravitational forces. The gravitational force between two objects depends on the distance between their centers. Let's denote the maximum gravitational force as Fmax and the minimum gravitational force as Fmin.
According to Newton's law of universal gravitation, the gravitational force (F) between two objects is given by the formula:
F = (G * M1 * M2) / r^2
where G is the gravitational constant, M1 and M2 are the masses of the two objects, and r is the distance between their centers.
Since the mass of the moon remains constant, the variation in gravitational force is due to the variation in the distance between the moon and the planet.
Let rmax be the maximum distance from the moon to the center of the planet, and rmin be the minimum distance. The relationship between Fmax and Fmin can be expressed as:
Fmax = (G * Mm * Mp) / rmin^2,
Fmin = (G * Mm * Mp) / rmax^2,
where Mm represents the mass of the moon, and Mp represents the mass of the planet.
Given that the maximum gravitational force exceeds the minimum gravitational force by 9.8%, we can write:
Fmax = Fmin + 0.098 * Fmin.
Substituting the expressions for Fmax and Fmin from above, we have:
(G * Mm * Mp) / rmin^2 = (G * Mm * Mp) / rmax^2 + 0.098 * (G * Mm * Mp) / rmax^2.
We can now cancel out the common factors:
1 / rmin^2 = 1 / rmax^2 + 0.098 / rmax^2.
Combine the fractions on the right-hand side:
1 / rmin^2 = (1 + 0.098) / rmax^2.
Simplify the left-hand side:
1 / rmin^2 = 1.098 / rmax^2.
Cross-multiply to obtain:
rmax^2 = rmin^2 / 1.098.
Take the square root of both sides:
rmax = sqrt(rmin^2 / 1.098).
Now we can find the ratio rmax/rmin by dividing rmax by rmin:
rmax/rmin = sqrt(rmin^2 / 1.098) / rmin.
Cancel out the common factors:
rmax/rmin = sqrt(1 / 1.098).
Evaluating this expression, we find:
rmax/rmin ≈ 0.953.
Therefore, the ratio rmax/rmin is approximately 0.953.
To find the ratio rmax/rmin, we need to determine the relationship between the maximum and minimum gravitational forces exerted on the moon by the planet.
We know that gravitational force is given by the equation:
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 (in this case, the moon and the planet),
and r is the distance between the centers of the two objects.
Since we are given that the maximum gravitational force exceeds the minimum gravitational force by 9.8%, we can set up the following equation:
Fmax = Fmin + 0.098 * Fmin
Simplifying the equation, we get:
Fmax = 1.098 * Fmin
Now, we can use the equation for gravitational force to express the distances in terms of the gravitational forces:
Fmax = G * (m1 * m2) / rmax^2
Fmin = G * (m1 * m2) / rmin^2
Substituting these expressions into the equation for the maximum gravitational force, we have:
1.098 * G * (m1 * m2) / rmin^2 = G * (m1 * m2) / rmax^2
Canceling out the masses and gravitational constant, we get:
1.098 / rmin^2 = 1 / rmax^2
Rearranging the equation, we have:
rmax^2 = rmin^2 / 1.098
Taking the square root of both sides:
rmax = sqrt(rmin^2 / 1.098)
Thus, the ratio rmax/rmin is:
rmax/rmin = sqrt(rmin^2 / 1.098) / rmin
Simplifying further, we have:
rmax/rmin ≈ 1 / sqrt(1.098)
Calculating this expression, we find:
rmax/rmin ≈ 0.945
Therefore, the ratio rmax/rmin is approximately 0.945.