Relevant equations:

Use the given values to solve for the force.

Newton's law of universal gravitation states:

We can write two equations for the gravity experienced before and after the doubling:

The equation for gravity after the doubling can be simplified:

Because the masses of the spheres remain the same, as does the universal gravitation constant, we can substitute the definition of Fg1 into that equation:

The the gravitational force decreases by a factor of 4 when the distance between the two spheres is doubled.

Newton's law of universal gravitation states that:

We can write two equations representing the force of gravity before and after the doubling of the mass:

The problem gives us  and we can assume that all other variables stay constant.

Substituting these defintions into the second equation:

This equation simplifies to:

Substituting the definition of Fg1, we see:

Thus the gravitational forces doubles when the mass of one object doubles.

When the elevator accelerates upward, we know that an object would appear heavier. The normal force is the sum of all the forces added up, and in this case it is . We know that the normal force has two components, a component from gravity, and a component from the acceleration of the elevator. Using this equation, we can determine the mass of the block, which doesn't change:

 is acceleration due to gravity and  is the acceleration of the elevator.

When substituting in the values, we get 

Solving for , we get 

Since the elevator is descending at constant speed, no additional force is applied, therefore the force that the spring scale reads is only due to gravity, which is calculated by:

To do this problem we have to realize that the force of gravity acting on the bullet is equal to the centripetal force. The equations for gravitational force and centripetal force are as follows:

If we set the two equations equal to each other, the small  (mass of the bullet) will cancel out and  will disappear from the right side of the equation.

 is the universal gravitational constant 

 is given to be  and  to be .

If we plug everything in, we get

 or 

We can use Newton's second law:

Set up equations for the force on the moon and the force on Earth:

Now we can use substitution:

From this, we can see that . Using the acceleration due to gravity on Earth, we can find the acceleration due to gravity on the moon.

The weight of the object on Earth's surface is:

The force on the new planet is five times that on Earth, so we can simply multiply:

We know that the weight of an object is given by:

 is the mass of the object and  is the gravitational acceleration of whatever planet the object happens to be on.

We know the gravity on the unkown planet, so the weight of the woman is given by:

We need only to find the mass of the woman to solve the problem. Since the mass of the woman is constant, we can use the information about her weight on Earth to figure out her mass.

Use this mass to solve for her weight on the new planet.

Escape velocity is the velocity at which the kinetic energy of an object in motion is equal in magnitude to the gravitational potential energy. This allows us to set two equations equal to each other: the equation for kinetic energy of an object in motion and the equation for gravitational potential energy. Therefore:

In this case,  . Substitute.

Since the radius of the new planet is  or  the radius of the original planet, and they have the same masses, we can equate:

Where  is the escape velocity of the larger planet and  is the escape velocity of the smaller planet.

Since the direction of the gravitational force is directly towards the center of the Sun, it lies in the plane of the comet's orbit. Since torque is , the torque is always zero. That is why the comet's angular momentum is conserved in orbit.