# High School Physics : Newton's Law of Universal Gravitation

## Example Questions

### Example Question #31 : Newton's Laws

An asteroid with a mass of  approaches the Earth. If they are  apart, what is the asteroid's resultant acceleration?

Explanation:

The relationship between force and acceleration is Newton's second law:

We know the mass, but we will need to find the force. For this calculation, use the law of universal gravitation:

We are given the value of each mass, the distance (radius), and the gravitational constant. Using these values, we can solve for the force of gravity.

Now that we know the force, we can use this value with the mass of the asteroid to find its acceleration.

### Example Question #32 : Newton's Laws

Two asteroids exert a gravitational force on one another. By what factor would this force change if one asteroid doubles in mass, the other asteroid triples in mass, and the distance between them is quadrupled?

Explanation:

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.

### Example Question #41 : Newton's Laws

satellite orbits  above the Earth. The satellite runs into another stationary satellite of equal mass and the two stick together. What is their resulting velocity?

Explanation:

We can use the conservation of momentum to solve. Since the satellites stick together, there is only one final velocity term.

We know the masses for both satellites are equal, and the second satellite is initially stationary.

Now we need to find the velocity of the first satellite. Since the satellite is in orbit (circular motion), we need to find the tangential velocity. We can do this by finding the centripetal acceleration from the centripetal force.

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 r 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 Fc=m∗ac. 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!

This value is the tangential velocity, or the initial velocity of the first satellite. We can plug this into the equation for conversation of momentum to solve for the final velocity of the two satellites.

### Example Question #42 : Newton's Laws

An astronaut lands on a planet with the same mass as Earth, but twice the radius. What will be the acceleration due to gravity on this planet, in terms of the acceleration due to gravity on Earth?

Explanation:

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 mg. 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 radius equal to twice that of Earth. That means it has a radius of 2r. It has the same mass as Earth, mE. Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

Expand this equation to compare it to the acceleration of gravity on Earth.

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 one quarter of what it would be on Earth.

### Example Question #43 : Newton's Laws

Two satellites are a distance r from each other in space. If one of the satellites has a mass of m and the other has a mass of 2m, which one will have the smaller acceleration?

They will both have the same acceleration

Neither will have an acceleration

We need to know the value of the masses to solve

Explanation:

The formula for force and acceleration is Newton's 2nd law: . We know the mass, but first we need to find the force:

For this equation, use the law of universal gravitation:

We know from the first equation that a force is a mass times an acceleration. That means we can rearrange the equation for universal gravitation to look a bit more like that first equation:

can turn into:  respectively.

We know that the forces will be equal, so set these two equations equal to each other:

The problem tells us that

Let's say that   to simplify.

As you can see, the acceleration for  is twice the acceleration for . Therefore the mass 2m will have the smaller acceleration.

### Example Question #44 : Newton's Laws

In the International Space Station, which orbits the Earth, astronauts experience apparent weightlessness for what reason?

The station is so far away from the center of the Earth

The astronauts and the station are in free fall toward the center of the Earth

The station’s high speed nullifies the effects of gravity

The station is kept in orbit by a centrifugal force that counteracts the Earth’s gravitational force

There is no gravity in space

The astronauts and the station are in free fall toward the center of the Earth

Explanation:

The space station and the astronauts inside are in a constant state of free fall toward the center of the Earth.  However, because they have such a high horizontal velocity and because the Earth is curved they will always be falling toward the earth as the Earth curves away from them.  IF the space station were to slow down, they would land on the Earth.  The high speed in the horizontal direction, keeps them in a parabolic flight path that aligns with the curvature of the Earth.

### Example Question #45 : Newton's Laws

An astronaut lands on a new planet. She knows her own mass, , and the radius of the planet, . What other value must she know in order to find the mass of the new planet?

The orbit of the planet

The density of the planet

The force of gravity she exerts on the planet

Air pressure on the planet

The planet's distance from Earth

The force of gravity she exerts on the planet

Explanation:

To find the relationship described in the question, we need to use the law of universal gravitation:

The question suggests that we know the radius and one of the masses, and asks us to solve for the other mass.

Since G is a constant, if we know the mass of the astronaut and the radius of the planet, all we need is the force due to gravity to solve for the mass of the planet. According to Newton's third law, the force of the planet on the astronaut will be equal and opposite to the force of the astronaut on the planet; thus, knowing her force on the planet will allows us to solve the equation.

### Example Question #46 : Newton's Laws

Two satellites in space, each with a mass of , are  apart from each other. What is the force of gravity between them?

Explanation:

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

We are given the constant, as well as the satellite masses and distance (radius). Using these values we can solve for the force.

### Example Question #47 : Newton's Laws

Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the Earth?  Which accelerates more?

Both the same; the Moon

The moon on the Earth; the Moon

The Moon on the Earth; the Earth

The Earth on the Moon; the Earth

The Earth on the Moon; the Moon