Remember that Newton's second law states that . The force acting upon the planet in question will be the force due to gravity. Once we find that, we can find the acceleration.

To solve, use Newton's law of universal gravitation:

Remember that the weight of the astronaut is the same as the gravitational force acting between the planet and the astronaut.

We are given the gravitational constant, the radius of the planet, the mass of the astronaut, and the magnitude of the force generated. Using these values in the universal gravitation equation, we can solve for the mass of the planet.

We are given the weight of a person on Earth in units of Newtons, which means we can recognize this as a force. The force that is acting on this person is the force due to gravity, which can be represented by the following equation:

 is the universal gravitational constant and is equal to 

 is the mass of object 1

 is the mass of object 2

 is the distance between the centers of the two objects

It's important to note that an object's mass will stay the same no matter where it is, but its weight will vary depending on where it is measured. Notice that when calculating the gravitational force, we need to consider the mass of two objects. If we set the mass of the Earth and the mass of the person in question as the two masses, we can rewrite the equation as:

To calculate how much the person weighs on the new planet, we need to consider the information given - that the new planet is twice as dense as Earth. This means that for a given volume, the new planet will have twice as much mass as Earth. Furthermore, we know that the mass of the person stays the same since, as mentioned above, mass is constant no matter where it is measured. And, if we are considering a case where the volume is the same, then the distance between the centers should also be the same. Thus, we can calculate the new force as:

This problem can be solved by looking at the formula for the gravitational constant, .

We are looking for the altitude above the Earth where the gravitation field is  in strength, so we proceed as follows

Use Newton's law of gravitation to make sense of this problem:

As you can see the force of gravity is inversely proportional to the square of the radius. Since  has a radius 6 times larger than , we state that  because the acceleration of the gravity of planet two divided by 36 would equal the smaller gravitational acceleration of planet one. 

Universal Gravitation law:

Assume  is the Moon, and is the object on the surface, then:

Will be analogous to in , which will be a negative number since it is pointing down.

Plug in values:

In an orbit, the centripital force will be equal to the gravitational force:

Where:

is the distance to the center of the earth from the center of the moon.

is the linear velocity of the moon

 is the mass of the moon

 is the mass of the earth

is the universal gravity constant,

Simplifying and plugging in values:

Solving for :

*note how the mass of the moon was irrelevant

Use the law of universal gravitation:

Convert  to and plug in values:

In an orbit, the centripetal force will be equal to the gravitational force:

Where:

is the distance to the moon's center. This will be the sum of the radius of the moon and the distance above the moon's center

is the linear velocity

is the mass of the ship

is the mass of the moon

is the universal gravity constant,

Simplify and plug in values:

Solve for :