It's difficult to make a rocket if you are one of the first rocket makers. Sometimes, you simply don't know the best way to design a rocket so you just have to pick something. This is exactly what happened with the early rockets. Robert Goddard's early design put the thruster at the very top of the rocket with the fuel tank at the bottom. The idea was that a top-mounted rocket would make the whole thing more stable. If the vehicle deviated from a perfect vertical motion, its lower center of mass would just make it swing back to the vertical position—just like a pendulum (a mass swinging on a string).
That's not what happens. A top-mounted thruster doesn't inherently make the rocket more stable and a bottom mounted rocket works just fine. Of course, you already know that you can put a rocket engine at the bottom of a rocket since just about every modern rocket does it this way. This idea that a top-mounted rocket makes the vehicle swing back and forth vertically is known as the pendulum rocket fallacy. Let's play around with some physics models so we can figure out exactly what's going on here.
I want to build a model that is as basic as possible—but can still show the main properties of a real rocket. It might seem silly, but here is my rocket design.
Yes, this rocket is just three masses connected by springs. Why? It's because this is the simplest design that would be a "mostly" rigid object but could still be modeled by calculating the forces on individual point masses. For each mass, there are three forces acting on it—the gravitational force and then the two spring forces. If I know the stiffness of the springs and the locations of all the masses, I can find these spring forces. Once I have the net force on each mass, I can use that to update the momentum and position of each mass after some very short time interval—then I can just keep repeating this process to get the motion of the whole rocket.
The cool thing about this method is that I only use simple forces on individual points, but from that I can get the motion of the entire rocket. Oh, I'm also going to plot the center of mass of this thing. OK, let's do this. I'm going to start off with my rocket as an actual pendulum. That means I will fix the position of the top mass and then just let the two bottom masses do whatever the forces tell them to do. But wait! Since this is a rocket, I need to hold it in place with some type of rocket engine. In this case, that rocket will push on the top mass in whatever direction (and magnitude) it needs to keep the thing still so it can act like a pendulum. It will essentially put the rocket into a hovering mode instead of taking-off mode. Here's what I get. Note: this is ACTUAL code. You can see and edit the code by clicking the pencil icon.
Yes, it does indeed swing back and forth—but what about that force? Since the body of the rocket is changing momentum while swinging, this causes the connecting springs to stretch and exert a changing force on the top mass. In order to keep that top mass stationary (for the whole rocket pendulum thing) the thrust force must also change both in magnitude and in direction of the thrust. So, this is not a normal rocket—but it is a fairly normal pendulum. It looks like things are working. That means we can go to the next version of the rocket.
What if we replace the thruster in the previous example with a more realistic rocket? This means that it will have a constant magnitude thrust and always point directly away from the center of mass for the rocket. So, if the rocket rotates, the thrust force will also point in a different direction—you know, like a real rocket engine.
Are you ready? Here is what that looks like (this is just a gif, but here is the code if you want to see it).
Notice that this NOT a pendulum. This is just a flying (and accelerating) rocket. What the heck is going on with this rocket? I've set the magnitude of the rocket thrust to just be a little bit more than the weight of the rocket so that it doesn't take off really fast. And we can figure this motion out by thinking about forces and torques. Forces tell us stuff about the motion of the center of mass of the rocket and torques tell us about the rotation about the center of mass. I'll start with the forces first.
As a rigid object, there are essentially just two forces acting on the rocket. There is the thrust force from the rocket engine (mounted at the top of the vehicle) and then there is the gravitational force. The gravitational force is an interaction between the Earth and every part of the rocket. However, no one wants to deal with a bunch of tiny gravitational forces exerted on every part of the rocket. Instead, you can replace all of these tiny gravitational forces with just one force—and that one force would act at a location called the center of mass. For this rocket, the center of mass is right in the middle of the object since the single top mass is equal to the sum of the two bottom masses (I made it that way).
Here is a diagram showing the forces on the rocket.
But what do these forces do? If the net force is zero (zero vector), then the object will either be at rest or move with a constant velocity. If the net force is not zero, then the object will have an acceleration. You can find the value of the acceleration using Newton's second law.
If you think of this thrust as a force (which it is), then there will be a component of this force in both the horizontal and vertical directions (since the rocket is tilted). Since there is a force in the horizontal direction the vehicle's horizontal velocity increases. The vertical component of the thrust is also slightly greater than the downward gravitational force so that it also increases in velocity (slightly) in the vertical direction. But the key thing here is that if there is a sideways force, the rocket will accelerate horizontally and not stay in the same position.
Now, what about the torque? My most basic explanation of torque is that it's like a rotational force. Net forces cause accelerations but net torques cause angular accelerations. So, if there is a net torque on an object it will rotate at an increasing rotation rate. But how do you find the torque? Imagine that you have a nut that you want to tighten using a wrench. In order to tighten it, you pull on the wrench in a direction perpendicular to the tool. Like this.
The torque in this case depends on three variables: the magnitude of the force (F), the distance between the force and the point about which you want to calculate the torque (often called the torque-arm, r) and the angle between the force and the torque-arm (θ). In the case above, the angle between the force and the torque-arm is 90 degrees. Since the sine of 90 is 1, this gives you the maximum torque for that force and torque-arm. If you need more torque, you could pull harder—or you could get a longer wrench with a greater torque-arm.
But what if you pulled with a force such that the angle was off the vertical axis? Like this.
If you want to tighten the bolt, this a bad idea. You get less torque at this angle (and you would pull the wrench off the nut). In fact, if you let the angle go to zero degrees, you get zero torque. So, if you imagine drawing a line through the force at the point of application and that line goes through your torque point (in this case, that's the nut), then the torque is zero. Remember, with zero torque you will get no change in rotational motion.
So, by mounting the rocket engine at the top of the rocket you get zero torque since a line through the force passes through the center of mass and the rocket doesn't swing back into a vertical position. But what is different with an actual pendulum? The key is the rotation point. For the free-flying rocket, it can rotate about its center of mass. Neither the gravitational force nor the rocket thrust force exerts a torque. However, when the top of the rocket is fixed in place (in that first pendulum example), the rocket must rotate about the top point. In this case, the gravitational force does indeed exert a torque and this is what causes it to swing back and forth.
OK, you should be able to predict what happens if I put the rocket thruster at the bottom of the vehicle. For this case, I'm just going to rotate the rocket all the way upside down so that the one single mass is now at the bottom. Here's what that looks like (and here is the code).
See. It still works. This shows the pendulum rocket fallacy. Putting the rocket engine at the top of the vehicle doesn't make it swing back to the vertical position, so there's no point in putting the engine up there. It makes much more sense to have the rocket at the bottom—you know, because all of that hot stuff that gets shot out of the thruster. If you have that at the top you are just going to damage your vehicle.
This isn't about rockets, this is about Iron Man. Actually, this is a response to some YouTube comments on my appearance on WIRED's Technique Critique in which I look at the physics of superhero movies. For one of the scenes, I looked at the way Iron Man flies (in the movies) using thrusters on both his feet and his hands. Yes, in the video I did indeed say "thrusters on the bottom of a rocket are a little problematic"—that's the exact same pendulum rocket fallacy that Goddard made with his first designs. Oops. It just feels like the bottom mounted rocket would be like holding up a vertical pencil from the bottom—but as you can see, that's not the case if the rocket is accelerating.
However, there are two things that are very different about Iron Man flying. First, in the scene from Iron Man (the movie), Tony Stark is hovering in a stationary position. This means that he is not accelerating and the total force must be zero (zero vector). Imagine that Iron Man is slightly leaning to the side. With just one thruster, you could not have a force that both pushed through the center of mass (zero torque) and have zero horizontal component. He would either accelerate to the side or change his rotational motion. So, let's say he wants to use a single thruster and remain stationary (while tilted). This is what it would look like.
Just to be clear, that uses a thruster force with a constant magnitude and direction—that's what you need to keep the center of mass stationary.
The second thing that makes Iron Man different than a rocket is that he doesn't have a single thruster on his feet—he has two (since he has two feet). This means the thruster forces do not have a zero torque arm and do indeed exert torques. Oh sure, it is indeed possible to have a zero torque situation with two feet thrusters, but let's imagine that he is tilted slightly off axis and only flying with thrusters on his feet. It might look something like this.
In this case, the total forces add up to zero (zero vector) so that he would remain stationary and hover. However, in order to get the total torque about the center of mass equal to zero, he would need to have a greater thrust for force F1 (on the left side). This greater F1 force would then provide the equal and opposite torque for thrust F2. It needs to be greater since its torque arm is shorter. But if you increase the force on F1, you would need to decrease the force on F2 in order to remain in a hovering position. It's a pretty difficult task.
If you can get the thrusters on your hands, this means that the torque arms for the two forces are greater (farther from the center of mass). This means that it's easier to make small angle adjustments to change the torque without changing the net force too much. But you don't have to imagine this situation—it's real.Yes, there is a real life flying suit that uses hand thrusters. It's created by Gravity Industries and it's really awesome.
OK, but what about the pendulum rocket fallacy? Let me summarize. The pendulum rocket fallacy is the incorrect idea that a top mounted rocket will make the vehicle more stable than a bottom mounted rocket. But yes, you can put rockets at the bottom of a rocket (if it's accelerating) and it will fly just fine. Yes, I said something not quite true in my superhero physics video. Finally, Iron Man is not a normal rocket.
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