For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a mass equal to six times that of Earth. That means it has a mass of . It also has four times the radius of Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

Expand this equation in order to combine the non-variable terms.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a mass equal to twice that of Earth. That means it has a mass of . It also has half the radius of Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

Expand this equation in order to combine the non-variable terms.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

The acceleration due to gravity on this new planet will be eight times what it would be on Earth.

For this comparison we can use the law of universal gravitation:

The astronaut notices that the force due to gravity on this planet is half what it is on Earth. We can thus relate the force equations for the two planets.

refers to any arbitrary mass on the surface of the planet, and will be constant. The radii of the two planets are equal, and the gravitational constant is also equal. These three variables can cancel out from both sides of the equation.

This relation shows us that the mass of the new planet will be half that of Earth.

For this comparison we can use the law of universal gravitation:

The astronaut notices that the force due to gravity on this planet is twice what it is on Earth. We can thus relate the force equations for the two planets.

refers to any arbitrary mass on the surface of the planet, and will be constant. The radii of the two planets are equal, and the gravitational constant is also equal. These three variables can cancel out from both sides of the equation.

This relation shows us that the mass of the new planet will be twice that of Earth.

The equation for the force of gravity between two objects is:

Using this equation, we can select arbitrary values for our original masses and distance. This will make it easier to solve when these values change.

is the gravitational constant. Now that we have a term for the initial force of gravity, we can use the changes from the question to find how the force changes.

We can use our first calculation to see the how the force has changed.

To solve this problem, use the law of universal gravitation.

Remember that  is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

This force will be the same for both objects. The Earth will exert the same force on the satellite as the satellite exerts on the Earth. This makes sense, given Newton's third law.

To solve this problem, use the law of universal gravitation.

Remember that  is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

This force will be the same for both objects. The Earth will exert the same force on the satellite as the satellite exerts on the Earth. This makes sense, given Newton's third law.

To solve this problem, first recognize that the force due to gravity of the Earth on the satellite is the same as the centripetal force acting on the satellite. That means .

Solve for  for the satellite. To do this, use the law of universal gravitation.

Remember that  is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

Now that we know the force, we can find the acceleration. Remember that centripetal force is . Set our two forces equal and solve for the centripetal acceleration.

Now we can find the tangential velocity, using the equation for centripetal acceleration. Again, remember that the radius is equal to the sum of the radius of the Earth and the height of the satellite!

To solve this problem, use the law of universal gravitation.

Remember that  is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the mass of the Earth, force of gravity, and distance to solve for the mass of the satellite.

To solve this problem, we need to set up the law of universal gravitation.

When the satellite was whole:

After the satellite breaks:

We can now compare these two equations. Start by expanding the second equation.

We can substitute our first equation into the second.

and 

This tells us that the final force of gravity is equal to one half the original force of gravity.