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 28.2 %. 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 first need to understand the relationship between the gravitational force and the distance of the moon from the planet.
Newton's law of universal gravitation states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Let's assume the mass of the moon remains constant. According to the problem, the maximum gravitational force exerted on the moon by the planet exceeds the minimum gravitational force by 28.2%. We can express this mathematically as:
Fmax = (1 + 0.282) * Fmin
Now, let's consider the relationship between the gravitational force and the distance from the center of the planet. The force is inversely proportional to the square of the distance, so we can write:
Fmax = G * (m1 * m2) / rmax²
Fmin = G * (m1 * m2) / rmin²
where G is the gravitational constant, m1 and m2 are the masses of the moon and planet, and rmax and rmin are the maximum and minimum distances from the center of the planet.
Since the masses of the moon and planet remain constant, we can cancel them out, giving us:
Fmax / Fmin = (rmin / rmax)²
Substituting the given relationship between Fmax and Fmin, we have:
(1 + 0.282) * Fmin / Fmin = (rmin / rmax)²
Simplifying, we find:
1 + 0.282 = (rmin / rmax)²
Taking the square root of both sides, we get:
√(1 + 0.282) = rmin / rmax
Finally, to find the value of rmax / rmin, we take the reciprocal of both sides:
1 / √(1 + 0.282) = rmax / rmin
Therefore, the ratio rmax / rmin is approximately 0.936, or rounded to three decimal places, 0.936.
Let's assume that the minimum gravitational force on the moon occurs when it is at the distance rmin from the center of the planet, and the maximum gravitational force occurs when it is at the distance rmax from the center of the planet.
According to Newton's law of gravitation, the gravitational force (F) between two objects is given by:
F = G * (m1 * m2) / r^2,
where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.
Here, we are considering the force exerted by the planet on the moon, so m1 is the mass of the planet, and m2 is the mass of the moon. Since the masses of the planet and the moon don't change, we can ignore them in our calculations.
Let's denote the minimum gravitational force as Fmin and the maximum gravitational force as Fmax. We are given that Fmax exceeds Fmin by 28.2%, which mathematically can be expressed as:
Fmax = Fmin + 0.282 * Fmin.
Now, we can set up an equation using the above expression and solve for the ratio rmax/rmin:
(G * m1 * m2) / rmax^2 = (G * m1 * m2) / rmin^2 + 0.282 * (G * m1 * m2) / rmin^2.
Cancelling out the G, m1, and m2 terms, we get:
1 / rmax^2 = 1 / rmin^2 + 0.282 / rmin^2.
Rearranging the equation, we have:
1 / rmax^2 - 1 / rmin^2 = 0.282 / rmin^2.
Combining the left side of the equation, we get:
(rmin^2 - rmax^2) / (rmin^2 * rmax^2) = 0.282 / rmin^2.
Multiplying through by rmin^2 * rmax^2, we obtain:
rmin^2 - rmax^2 = 0.282 * rmax^2.
Simplifying, we have:
rmin^2 - rmax^2 = 0.282 * rmax^2.
Combining like terms, we get:
0.718 * rmax^2 = rmin^2.
Taking the square root of both sides, we have:
sqrt(0.718) * rmax = rmin.
Now we can find the ratio rmax/rmin:
rmax / rmin = rmax / (sqrt(0.718) * rmax) = 1 / sqrt(0.718).
Using a calculator, we find that the ratio rmax/rmin is approximately 1.0698.
Therefore, the ratio rmax/rmin is approximately 1.0698.
max gravitational force=1.282 x min gravitational force. therefore, rmax^2/rmin^2=1.282
square root both sides