Superposition of Gravitational Forces in <cite>Angry Birds Space</cite>

No. This is not a Lagrange Point in Angry Birds Space. It is important, though.

Let me get this over with. Why is this not a Lagrange Point?

What is a Lagrange Point?

Basically, a Lagrange point is where an object can appear to remain stationary relative to some other object due to the sum of gravitational forces from two large objects.

I know that definition sort of sucks. Let me instead show you the L2 Lagrange point. Start with the Earth orbiting the Sun.

There is essentially only one force on the Earth, the gravitational force from the Sun. This force causes the Earth to have a centripetal acceleration. In the direction of the Sun (the radial direction), I can write:

Basically, the centripetal acceleration depends on r and so does the gravitational force. The result is that for a circular orbit, a particular radius orbit would have a particular angular velocity.

So, what if I want to place a space station such that it stays in the same relative location to the Earth-Sun system? Well, if it is farther away from the Sun than the Earth it would a lower angular speed. I could make it have the same speed, but it would need a greater gravitational force on it than just from the Sun. BOOM! It just so happens that I can place this space station in a location where there are TWO gravitational forces on it.

With both of these forces in the same direction, it is enough to get the space station to have the same angular velocity as the Earth. And this is a Lagrange point. Fun, right? But you know what? I want to look at Angry Birds instead. Why is this Angry Birds case not a Lagrange point? Basically because the two gravitational object (the asteroids) aren't even moving. So, it is not the same thing. I guess you could say it is sort of like a Lagrange point -- I could live with that. As long as we all understand that it isn't really. But I guess if the two asteroids were in that position, they would attract each other - that is unless they were orbiting each other. But in that case, we would have a non-inertial reference frame and there would be some fake forces added in.

Sum of Gravitational Forces

Now for some analysis. Remember from my previous analysis, I found that there were essentially three things for a bird in the gravitational influence of a rock:

• A constant gravitational force. For the previous case it was (30 m/s²)m (where m is the mass of the bird) and in a direction toward the center of the rock.
• A constant frictional force. The value before was (30 m/s²)m in the opposite direction as the velocity of the bird.
• Some type of speed limit. The bird can only go up to a speed of 30 m/s.

I really don't know if these values are the same for every level, but I am going to take a guess that at least the gravitational force is still constant. The video above seems to suggest that it is indeed constant in magnitude. Why? Because the bird can be in a stable oscillation. Here is a diagram stuck in the two gravitational fields (OK, you win -- I will call it a Lagrange point to make you happy).

I picked a point where the bird was stopped for an instant. I assume the frictional force is zero here - but I am really not sure. For these two forces, the net force would be to the left. Of course, if this was a 1/r² gravitational force the forces could still do this. The problem is that with a slight deviation, one would be greater in magnitude than the other one. This would cause the bird to not stay in the same path.

So, here is the question: Can I model this Lagrange-point oscillation to get an estimate for the gravitational force? I can try at least.

Let me call the point right in the middle of the two rocks the origin and the location of the bird, x. If the centers of the two rocks are a distance R away, then I can draw this:

The component of the gravitational force in the y-direction will cancel with the other gravitational force. The x-component of this one gravitational force will be:

If the two gravitational forces have the same magnitude,the total force on the oscillating bird would just be twice this value. Notice that this is close, but not exactly the same, as simple harmonic motion. If you have a force that is proportional to x, this would be just like a spring. Either way, this won't stop me from modeling the motion of an object with this force. I would go ahead and model this motion, but I need to get some initial conditions from the video. Might as well start with the actual data.

Video Analysis

Here is a plot of one of the oscillating Angry Birds as a function of time.

I will be honest. This isn't what I expected. It seems odd that it will go to a greater x-value than the last oscillation. Well, there isn't much to do about but to see if I can model the motion. Let me put the origin in between the two asteroids with a bird starting from rest at x = -3.89 meters (of course, using the slingshot scale of 4.9 meters). Also, I will assume the gravitational field has a constant magnitude of 30 N/kg (as I found in another level).

Here is my first model without the frictional force. The blue line is the model and the green is the data from Angry Birds Space.

Close, but not close enough. Let me add in the frictional acceleration of 3 m/s². Here is the new plot.

Clearly, that didn't work either. The frictional force just stopped it too soon. I could lower the friction to make it look a little bit better, but it is always going to be moving toward a smaller and smaller amplitude. This is odd. It almost looks like this is the sum of two slightly different oscillations which would give beat frequencies. OK, this is crazy. What if I look at the acceleration of the bird when it is stopping? It looks like for all of these turning around points the acceleration is about the same:

They all give a value of around 6 m/s². What if I use this acceleration to get an estimate for the gravitational force on the birds? If I use an x value of 3.5 and an R of 11 meters, then the magnitude of the force from each asteroid would be 9.8 Newtons (I put the mass of the bird as 1 kg for simplicity). OK. Let me change the force in my numerical calculation from 30 Newtons to 9.8 Newtons (and remove the friction).

OK. That looks nice. Let me see if I can add friction back in. It obviously isn't going to be near the 3 Newtons from my previous study. This is the best I could get. I put the gravitational force at 10 Newtons and the frictional force at 0.1 Newtons.

I think that is the best I am going to get. I suspect there is something still not correct. Either the actual Angry Birds Space game has a rounding error, or the friction force they use is weird. Oh, maybe the two gravitational forces from the two rocks have different values. It doesn't matter too much. I think this shows that you can get an oscillation with a constant magnitude of gravitational force. However, what about the strength of this force? It is clearly different than the other rock I looked at. Let me see if I can match the motion of a bird with plain orbital-type motion under the influence of just one of the rocks.

Here is a plot of actual Angry Birds Space data from that level and a model. In this model, I have the gravitational field at 60 N/kg and a frictional acceleration of 3 m/s² (just like before for the friction.

It doesn't fit quite as well as I would like. However, I can say with a fair amount of certainty that the gravity for this level has a different value than in the previous level.

Conclusion

Really, I am a little disappointed. I thought I would look at this oscillation data as further evidence of my previous Angry Birds forces model. Well, that doesn't seem true. Here is what I have:

• If the gravitational forces add together in the overlap region, a theoretical model would have the bird oscillated. This mostly agrees with experimental evidence.
• In order to get the model to agree with oscillation data, each rock would have a gravitational field of about 10 N/kg with a very small frictional acceleration of about 0.1 m/s². This is different than the gravitational field and acceleration of the previous level I looked at which had g = 30 N/kg and a = 3 m/s².
• Although I used a gravitational field of 10 N/kg in the overlap area (for each rock), I had to use a value of 60 N/kg for a bird in motion around just one asteroid. Odd.
• There is some weird oscillation in the overlap area. The bird's amplitude of oscillation gets a little bit bigger before it gets smaller.
• I have this feeling that developers at Rovio (the creators of Angry Birds) are putting in these seemingly random forces to prevent me from figuring things out.

Clearly, more work needs to be done in the field of High Energy Angry Birds. Oh, and I am sure I will get the comment: Why are you wasting your time on this? For me, this analysis is the REAL Angry Birds game.