Question

A planet has two moons with identical mass. Moon 1 is in a circular orbit of radius r. Moon 2 is in a circular orbit of radius 2r. The magnitude of the gravitational force exerted by the planet on Moon 2 is which of the following compared with the gravitational force exerted by the planet on Moon 1?

half as large
twice as large
one-fourth as large
four times as large
the same

Answer 1

The magnitude of the gravitational force exerted by the planet on Moon 2 is one-fourth as large as the gravitational force exerted by the planet on Moon 1.

Step-by-step explanation:

The magnitude of the gravitational force exerted by a planet on an object depends on the object's mass and the distance between them, as described by Newton's Law of Universal Gravitation. The formula to calculate the gravitational force is 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 them

In this case, Moon 2 is in a circular orbit of radius 2r compared to Moon 1's orbit of radius r. The distance between the planet and Moon 2 is twice the distance between the planet and Moon 1. Since the gravitational force is inversely proportional to the square of the distance, the magnitude of the gravitational force exerted by the planet on Moon 2 is one-fourth as large as the force exerted on Moon 1

Answer 2

Answer: Half as large

Step-by-step explanation:

Newton's law of universal gravitation for Moon 1 is:

`F_(1)=G(Mm)/(r^(2))` (1)

And for Moon 2:

`F_(2)=G(Mm)/((2r)^(2))` (2)

Taking into account the mass of Moon 1 is equal to the mass of Moon 2

Where:

`F_(1)` is the gravitational force exerted by the planet on Moon 1

`F_(2)` is the gravitational force exerted by the planet on Moon 2

`G=6.674(10)^(-11)(m^(3))/(kgs^(2))` is the Gravitational Constant

`M` is the mass of the planet

`m` is the mass of each Moon

`r` is the distance between the planet and Moon 1

`2r` is the distance between the planet and Moon 2

Dividing (2) by (1):

`(F_(2))/(F_(1))=(G(Mm)/((2r)^(2)))/(G(Mm)/((r)^(2)))`

`(F_(2))/(F_(1))=(1)/(2)`

Isolating
`F_(2)`:

`F_(2)=\frac{1}(2) F_(1)`