I'm not sure what's happening in this trailer for Independence Day: Resurgence, but here's what I think: The aliens are back and not at all happy about losing the battle on Earth. As part of their plan for revenge, they deploy a large ship near the planet's surface. The ship is so massive that it gravitationally pulls objects (like buildings) toward it. Again, this is just my speculation from the video.

What kind of mass would a spacecraft need to pull things off Earth? Let's take a look at gravity and then make an estimate.

The Gravitational Force

People often think of gravity as "that thing that makes apples fall" or perhaps the reason you fell off your bike. Yes, this is the gravitational interaction, but there is so much more to it than that.

Scientists model gravity on Earth's surface as a downward force that is proportional to the mass of an object. The equation can be written as:

You probably don't like seeing this as a vector equation, but the vector part is important. It shows that both the force and g are vectors where g should be called the gravitational field. But forces don't come individually. Forces are an interaction between two objects. If the Earth pulls down on a person, the person also pulls up on the Earth.

But if a human exerts a gravitational force on the Earth, does a human also exert a force on another human? Yes. The gravitational force is an attractive interaction between any two objects with mass. We don't normally notice these attractive forces because the magnitude is tiny. However, there is an experiment that allows you to measure these forces.

This is a picture of a Cavendish torsion balance. It's named for Henry Cavendish, who used it to determine the gravitational constant.

The idea is to place small masses on a bar suspended by wire. The bar and balls mostly rotate freely. Place two large masses near them, and the gravitational force is strong enough to move the bar, twisting the wire. The amount of twisting is related to the gravitational force between these masses. The magnitude of this force can be written like this:

In this equation, we have:

G—the gravitational constant. This has a value of 6.67 x 10 -11 N*m 2 /kg 2 .

N*m /kg . m 1 and m 2 are the masses of the two interacting objects.

and m are the masses of the two interacting objects. r is the distance between the two objects. Hopefully the distance is much greater than the size of the objects so you can just use the center-to-center distance.

Since the value of G is so small, the attractive forces between normal objects (like humans) is insignificant.

But what about the constant gravitational force and the gravitational field g? This is the same thing as the universal gravitational force—just between an object and the Earth. If you put in the mass of the Earth (5.972 x 1024 kg) and the radius of the Earth for the distance between the objects (6.371 x 106 m) you get a force of 9.8 Newtons per kilogram—yup, just like g. Also, if you move 1,000 meters off the surface of the Earth, you will increase the distance between the object and the center of the Earth by 1,000 meters. But that's still 6.372 x 106 meters—just about the same as before. Since the radius of the Earth is so huge, the gravitational force doesn't seem to change with height (even though it really does).

The Gravitational Force from a Spaceship

What about the scene in Independence Day: Resurgence? Why would these buildings get pulled from the surface of the Earth? First, let's start with a normal building on a normal non-alien invasion day on the surface of the Earth. I'll assume there is nothing holding the building down beyond gravitational force (which is unlikely due to building codes).

These forces are balanced and the building is at rest. Of course balanced forces also could mean the object is moving at a constant speed, but if the object moves up it will lose contact with the ground and there will no longer be a force pushing up. If the building moves down, the ground force will increase (like a spring) and push harder on the building. The only option is for the building to be at rest.

Now let's put a big spaceship (with a super-large mass) overhead.

In order for the building to be lifted, the gravitational attraction to the spacechip must be at least as large as that of the Earth. Sure, the spaceship is closer but it's going to have to be massive to have a significant effect. Now for some wild estimates. We don't really get a good view of this spacecraft, so I'm going to guess it is 5,000 meters above the surface of the Earth (it's probably much higher if it is really super big). In this case, I can solve for the mass of spacecraft by setting the two gravitational forces on a building equal to each other.

Putting in my values for the mass of the Earth (m E ), height of spacecraft (h), and radius of the Earth (R E ) I get a spaceship mass of 3.7 x 1018 kg. Just for comparison, this is about the mass of many of the large asteroids with a radius of around 70 km. Of course this spaceship could be even smaller if it had a higher density. Oh, and I'm not removing the possibility that there is something other than just a gravitational force due to mass of the spaceship. Perhaps the aliens have a technology that allows them to create gravitational fields using something other than mass.

Homework

Here are some homework questions for you.