In the first part, we examined some of the physics behind getting energy into an ax and how to estimate how much energy it will deliver and how fast it will travel. In this part, we're going to look at some design considerations that need to be respected when building an ax weapon.

I want a 50lb hammer that swings 1000mph. Can it be done?

So, you want a really big hammer that goes really really fast? Well that was what was suggested from part 1. Now it's time for a dose of reality. A 50lb hammer traveling 1000 mph can't be done given the weight limits imposed on our bots. Even the super-heavyweights would find such a hammer impossible to achieve. To explain why this is the case, let's examine a new physical concept called 'centrifugal force'.

You've probably heard of centrifugal force. I'd be willing to bet you've experienced it. Centrifugal force is that mysterious force that makes you slide across your seat when you're riding in a car that turns suddenly. The reason this force occurs has to due with one of Newton's laws of motion. In particular, Newton's law tells us that "an object in motion wants to stay in motion, likewise an object at rest wants to stay at rest". This is the basis for the concept we call inertia. We can restate this law in more practical terms as follows: It takes an outside force to change the direction and/or the speed of an object. When a force is applied in such a way as to make an object follow a circular path, we call that force a centrifugal force. In the case of our car, the tire's friction against the road provide the needed centrifugal force needed to allow the car to go around a corner. (They are also providing the angular acceleration needed to keep the car pointed in the right direction as well.) Now think about our ax head swinging at a high rate of speed. It's moving in a circle. Therefore, something must be providing the centrifugal force needed to make it do this. You've probably guessed where it's coming from. The ax handle of course. The handle (or 'arm' since we don't actually handle this ax) is providing the force needed to make the ax head go in a circle.

So, can't we just make a strong arm for our ax and swing it as fast as we like? Nope. The reason why is explained by another of one of Newton's laws which states: "For every action there is an equal and opposite reaction". Well that's clear as mud. What does that mean to us? It tells us that if the arm is providing a force to the ax head, something must pulling the arm in the exact opposite direction. That something is the weight of the bot sitting on the floor. So think about this. When the ax is passing directly overhead, the centrifugal force provided by the arm is pulling the ax head straight down to keep the ax moving in desired circular path. This means that the arm is both pulling down on the ax head and pulling up on the bot with an equal force, in effect trying to lift the bot off the floor.

"So what? My bot weights over 200lbs. How is an 8lb ax going to lift that much weight?", you may ask. I'll tell you why (I'll offer proof a little later). If we choose to build the ax described in part 1 (an 8lb ax on a 30 inch arm moving with 500 lb-ft of energy), the vertical lift force is approximately 240 pounds, which exceeds the maximum weight allowed for a heavyweight bot. In other words, the ax described in part 1 will cause the bot to leave the ground when fired. (It will actually somersault through the air.) It gets even better. If we reduce the energy so that the lift is less the than the weight of our bot, the bot will still try to slide across the arena floor because there is also a horizontal force being exerted as the ax moves from the 12 o'clock position to the 3 o'clock position. What's worse is that the horizontal force is even bigger than the vertical force because the ax has likely gained more speed since leaving the 12 o'clock position. (Our cylinder has been pushing against it the entire time.) In fact, the horizontal force is about twice that of the vertical force. There is simply not enough traction available from the tires to keep the bot from sliding. (I should state some assumptions that were made in the scenario just described. First, I'm assuming that the pneumatic actuator applies force to the ax during a full 180 degree swing. Second, I'm assuming the coefficient of friction of a rubber tire against a metal floor is 1.0.) So, the bottom line is that it is very unlikely that you'll ever see a heavyweight bot swinging an ax with more than 500 lb-ft of energy if we expect it to stay on the floor, even if we allow it to slide sideways. If we address the horizontal slide problem, then the maximum energy will be around 250 lb-ft. Being a super-heavyweight doesn't get you off the hook either. It's maximum energy is about 350 lb-ft. And before you run off and say you'll just change the weight or length of the ax to get around this, well surprise! It won't work. Even if we cut the weight of the ax by half, we still get the same numbers (or at least they're very close). Why? In order to keep the same energy with a lighter ax, it has to move faster which in turn produces the same centrifugal we had before. Changing the arm length can help a little, but physical space limitations don't allow a whole lot of options here either. (Doubling the length of the arm cuts the centrifugal force by half. But a 5 foot swing arm would be a challenge to operate.)

Oh yeah? Prove it!

I suppose I need to prove my assertion made in the previous section that you can't build a heavyweight bot with more than 250 lb-ft of energy in its ax weapon. After all, this does directly challenge claims made by some bot teams. Fair enough. (There is a way to violate this rule. I'll explain that trick later.)

In order to prove my case, I need to introduce another equation which tells us how much centrifugal force there is for a given mass moving in a circle of some arbitrary radius. The equation looks like this:

As you can see, the centrifugal force builds with the square of the angular velocity. This helps to explain why using a lighter ax head moving at a higher speed doesn't offer a solution to our problem of keeping the bot on the floor during weapon deployment.

The next thing we need is an equation which can tell us the angular velocity at some arbitrary point in the ax's movement. We need to make two assumptions to make this a bit easier. First, we'll assume the ax moves in a 180 degree arc. Second we'll assume that there is a direct relation to how far the ax moves versus the motion of the piston in the pneumatic cylinder. In other words, the ax completes half its swing when the piston has moved half its total travel and so on. Based on the second assumption, we can say that the energy in the ax has a linear relationship to its travel. This is a direct result of how work is defined. Remember from part 1 we learned that work is a force multiplied by the distance traveled. So when the piston moves half way it has supplied half the total energy to the ax and so on. Since we said the ax has a direct relation to the movement of the piston, it also follows the same rule. With this knowledge, we can define the energy in the ax at an arbitrary angle:

Now we need to know the angular velocity using an arbitrary energy. We already know this equation from part 1:

If we combine all these equations into big one we get:

Grit your teeth, there's two more equations we need. The last equation tells us how much centrifugal force there is for a given ax position, but it doesn't tell us which way it's pulling. Using the same units as before, the two new equations are:

Whew! A whole lot of voodoo just happened. But we now know a whole lot about what's going to happen when we swing our ax. Let's use our new equations one at a time. First we'll examine the equation which yields F_v. This is the portion of the centrifugal force that is trying to lift our bot from the floor. All we need to do is graph this equation and see if during any portion of the swing, the resulting force is greater than the weight of our bot. (I'll even save you some trouble. The greatest lift occurs when 'a' is about 115 degrees.)

The second equation (F_h) tells us the horizontal portion of the centrifugal force which is trying to drag us across the floor. We do need a graph for this one. To use it, you need to plot two things. The first is the weight of the bot minus the vertical lift given by equation for F_v. This tells us how much weight is on the wheels. (Traction is directly proportional to weight.) Next, plot equation F_h. If F_h exceeds the value obtained from the weight of the bot minus F_v, then the bot is going skiing. (Note: use the absolute value for F_h. We don't care which way it's going to slide, only if it's going to slide.) Unfortunately, this equation doesn't allow us to just solve for one value of 'a' and call it a day. However, as a rule of thumb, you can solve F_h at 180 degrees and be reasonably certain that if the result is less than the weight of the bot, then you're not going to have problems.

I rest my case.