I can't help myself. I have to say something about this awesome water slide as seen on io9.
You really should check out the io9 article - an interesting read. But for me, let me see if I can estimate what it would feel like to go through this crazy thing. To start, all I really have is the photo and a claim that the loop was about 15 to 20 feet high.
How would you model this crazy slide? Let me break this into two parts. Part 1 is the straight tube. During this part, the force diagram would look like this:
Since I am looking for the speed after it goes a certain distance, the best bet is to use the Work-Energy principle. If I take the person plus the Earth as the system, then I will still have the frictional force doing work as it slides down. Let me call the length of the slide s. This makes the work-energy principle like this:
In order to find the speed at the bottom, I will need to first find a value for the frictional force. Looking back at the force diagram, the forces in the direction perpendicular to the slide must add up to zero since the person doesn't accelerate that way. Along with this, I can use the model for friction that says it is proportional to the normal force.
I am not worried about the mass (it won't matter in the end), but I do need a value for the coefficient of kinetic friction. Since I have no actual data from this slide, I will have to look at something similar. Here is an older post with an analysis of a different slide. These are those big slides at the fair where you get on a potato sack or something. From that, I found a coefficient of kinetic friction with a value of 0.31. Let me just assume the water slide is a little bit less. How about 0.2? Everyone happy with that?
Now, if I assume the slider person starts from rest at the top of the slide, I can find out how find the slider would be moving just before entering the loop.
Actually, this is a little silly. I have both the length (s) and the height (h), but I could get a relationship between them from the angle of incline. Oh well.
What about the loop part? The force diagram would look similar, but I will draw it anyway.
An object moving in a vertical circle. Seems simple, right? You see problems like this in introductory physics. Or do you? No. You don't. You see a problem that asks about the forces at either the top or bottom of the circle. They never ask about the motion all the way around. It isn't so simple. The main problem is the force the tube exerts on the rider (normal force). This is considered to be a "constraint force". This means that the normal force exerts whatever force is necessary (up to its breaking point) to keep the rider from going past the tube. It constrains the motion of the person to the surface. Get it? Constraint force.
But then how do we deal with this force? A simple numerical model won't work. The main process in these numerical calculations is to do the following:
- For each small step in time:
- Calculate the total force.
- Use the total force to determine the change in momentum and thus the new momentum.
- Use the momentum to find the change in position.
- Rinse and repeat.
This method works well if I can find the forces based on position (like a spring) or velocity (like air resistance). However, the normal force doesn't depend on these things. What to do? Cheat. Well, not really cheat. Just sort of cheat. Here is the plan. First, I will assume the trajectory is in the path of a circle. From this I can calculate the acceleration in the direction towards the center of the circle based on the velocity and the radius.
This radial acceleration is due to two forces: the normal force (which is in the same direction as the radial acceleration) and a component of the gravitational force. Since I know the acceleration in the radial direction and the gravitational force, I can solve for the unknown normal force. The direction of this normal force will be towards the center of the circle.
With the normal force, I can then find the friction force. As a vector, it would be:
Here the "v-hat" is a unit vector in the direction of the velocity. But the point is that now I know all three vector forces (gravity, friction, and the normal force). From here, I can use the usual numerical model.
Apparent Weight
The first question that comes to my mind: what kind of forces would you feel if you make it around the loop? Ok, I first need to determine the starting height. If I assume a loop diameter of 20 feet (6.1 meters), a measurement of the image shows the starting height would be about 16.2 meters above the bottom of the loop. This would put the speed entering the loop at 15 m/s (33.5 mph).
This is bad. Why? Here is a quick animation of the loop if the starting speed is 15 m/s.
Yep, that's right. In this case the slider didn't make it around the top of the loop. Good thing they put that escape hatch in the tube. I guess my value for the coefficient of friction was too high. There is that water sliding down with you after all. If I change the coefficient of kinetic friction to 0.1, then the speed entering the loop would be 16.5 m/s and the slider would make it over the top.
Oh, you might notice that my animation included vectors representing the three forces. Notice two things about the normal force (white vector). First, it gets pretty huge. Second, in the case where the slider goes back down the direction of the normal force changed. This means that in order to stay on that circle, the tube would have to pull on the person. Of course that wouldn't actually happen. Instead, the slider would fall and crash into the top of the tube at a lower point. Ouch.
What if I want to plot the apparent weight. Remember that what you feel is not the gravitational force but instead all the other forces (because gravity pulls the same on all parts of you). I am pretty sure the apparent weight would be the sum of the frictional and normal forces. Here is a plot as function of time.
Wow. 10 g's when the slider first enters the loop? That seems crazy high. Let's just check. Just the normal force would be easy to calculate. If the slider is at the bottom of the loop going 16 m/s, then the following must be true for the forces in the y-direction (at that instant):
With a radius of 3 meters, this gives an acceleration of 10.2 g's. Wow. That is just crazy. If you are going any slower, you wouldn't make it over the loop. Any faster and you might die from the massive acceleration.
Changing the Coefficient of Friction
With the parameters as they are, what is the maximum value of the coefficient of friction for which you can get over the loop? Here is a plot of the maximum height in the loop for different starting values of μ.
What does this say? This says that if the coefficient of friction is less than around 0.18, you will make it to the top. Making it to the top and making it around the loop are two different things. If you just barely make it to the top, you will be there with a speed of zero. This means you wouldn't be moving in a circle. You would just fall straight down. In order to still be moving in a circle of radius R, the lowest speed would have no normal force pushing on you. This means that in the y direction we would have:
With a radius of about 3 meters, this would be a minimum speed of 5.4 m/s. Here is a plot showing the maximum height along with the speed at that height.
Here, the green line represents the speed and the horizontal red line shows the speed value of 5.4 m/s. From this, you would need a maximum coefficient of friction of 0.15 in order to just barely make it over the loop without crashing.
