The Physics of Scooting
04/05/08 21:00 Filed in: Physics
There you are, in your
rolling chair. You need that extra pencil, it is just
a few feet away but out of reach. For some reason,
you don't want to touch your feet to the floor. How
do you get that pencil? You scoot. Still don't get
it? Here is an example:
Why do people scoot? I don't know. My kids do it too. When does a human learn to scoot? How does a scoot work? Really, I am just going to answer that last question, but the others are interesting also.
Why do people scoot? I don't know. My kids do it too. When does a human learn to scoot? How does a scoot work? Really, I am just going to answer that last question, but the others are interesting also.
What is the big deal?
Don't you just move your head, and that makes you
move? Well, not really. Let me start from the
beginning - that is my favorite place. What if there
were no friction and at all and you started rocking
back and forth, what would happen? The simplest way
to explain this motion is to say that the center of
mass of the system remains constant. Newton's second
law of motion can be generalized for a system of
particles as:
Where ptotal is the momentum of all the particles the object is made up of. You could also say:
This is true, even if you don't believe me. I would happily derive this for you, but I like to keep my posts to the point.
So, in order to move, you have to change your momentum (from zero to something). To do this, you need an external force (so not you nor the chair). There are two external forces, but they cancel (gravity and the force of the floor pushing up). The result is that rocking back and forth will move the chair, but the chair will move right back when you return to your initial configuration.
To demonstrate this, I created a calculation in Vpython. This calculation has two masses, one on top of the other while on a floor. There is some driving force between the two masses to simulate the rocking motion. For this first case, the top mass only oscillates horizontally (although added vertical motion would not essentially change the results when no friction is present).
Below is a graph of the horizontal position of the top and bottom mass. For this calculation, the bottom mass is four times the top mass. (I also plotted the center of mass - the black line)
In case you are still having a difficult time picturing what is happening, here is a screen shot of the setup:
Basically, the red ball is the
top mass and the yellow block is the chair.
Note in the graph above, the center of mass does not change.
Well, this is easy to fix, an external force can be added with friction. Ok - I will repeat the above calculation WITH friction. (by the way, I also measured the coefficient of kinetic friction for a rolling chair and found it to be around 0.05 - I will post this analysis later).
Notice that now, the center of mass does indeed move. However, you never really go anywhere. The average displacement is still zero. This is because the same friction force that is used to move you forward is also there when you return your head to its original position. If you want scooting to work this way, you would have to not decrease the momentum of your head (and thus you would not move back). Of course this would probably mean that your head is no longer attached to your body - generally, a bad thing.
Wait, what if you moved fast one way, and slowly back. The only way this would matter is if the static friction force is very high (I will deal with that later). For a rolling chair, this motion would still leave you with an average displacement of the center of mass at zero.
The KEY to scooting
Ok, I am ready to reveal the real key to scooting. I hope I don't offend the entertainers that rely on the mystery of scooting as part of their act - but I must do this. We can only have a free society if the secret of the scoot is understood by all.
HERE it is: Make the frictional force different for different directions. To do this, you need to know a little something about friction. Friction is actually uber-complicated, but there is a simple model that works pretty well. It says (for sliding friction):
Here N is the force the floor
pushes on the object and μ is the coefficient of
friction. So, the harder you push two surfaces
together, the more friction there is. Hey wait!
This is a rolling chair, not a sliding block.
Well, this model still workings. This is sliding
friction, with friction in the axle.
So, one way to reduce the frictional force is to reduce the force between the floor and the chair. This can be accomplished if the top mass (head) is accelerating down. This motion would cause the chair mass to accelerate up so that the center of mass stays constant. This motion can be represented with the following:
(I know this picture looks a little crazy, but let me try to explain).
The key difference is that on the left, the top object is accelerating down and to the left. The right top object is accelerating up and to the right. The vertical acceleration is very important. For the picture on the right, the top mass is accelerating down. This means that a downward force must be exerted on the top mass (by the bottom mass - chair). According to Newton's 3rd law, there must be an equal force, but opposite direction on the bottom mass. I have represented this force by the red arrow labeled "top" (sorry for including both acceleration arrows and force arrows, they are not the same thing).
Ok, so the bottom mass does not move in the vertical direction. This means that all the vertical forces must add up to zero. In this case, that is gravity, the floor and the force from the top mass. Since the top mass is pulling up, this means that the floor does not need to push as much. With a reduced force from the floor, there is a smaller friction force.
For the picture on the right, the opposite happens. The top mass is accelerating up. This causes a downward force on the bottom mass leading to increased force from the floor. Increased force from the floor means a greater frictional force.
That may be a little confusing still, let me summarize.
So, what happens if I include both horizontal AND vertical motion in my model? In this case, I created a force between the top and bottom mass that looked like this (on the top mass)
Note that the amplitude of the
force in the y direction is 2 times that in the
x, this is just because it gives better results.
(also note that the force on the bottom mass
will be the negative of this force).
I know you are waiting for the results, you have been very patient. Here is a plot of the motion of the top and bottom mass when vertical motion is also included.
I have also displayed the frictional force (purple line). When the horizontal acceleration of the top mass is the greatest negative value (greatest negative curvature) the friction is minimized. When the horizontal acceleration of the top mass has the greatest positive value, the frictional force is a maximum. This produces a steady net motion of the center of mass (as well as the position of the chair). Note that this is likely not the most optimal accelerations of the top mass to produce the most efficient movement, but it is the first thing I tried.
Real Data
Below is data from the posted video above (from a student, who I will call Mad Mursavich in case he doesn't want me to post his real name). In this graph, I have plotted the horizontal position of the head and the chair as well as the vertical acceleration of the head. This vertical acceleration was obtained by differentiating the y position of the head twice (I used Plot to do this - an awesome FREE program)
Really, I would like to plot the frictional force as I did with the numerical model, but I don't have that data. What I do have is the vertical acceleration of the head. When the head has a positive acceleration, the head will be pushing down on the chair creating a greater frictional force. When the head acceleration is negative, this will reduce the frictional force. So you see that the chair is moving forward when the head acceleration is negative (lower frictional force). Note: I scaled the acceleration data so it would fit on the same graph. Sorry, but I am not yet an expert with Plot to make this look perfect.
I guess that is it, the physics of the scoot. I know I have more to do, and I will keep you updated. I feel like the Penn and Teller of scooting now - revealing the secrets. Now that you know the secret remember that with great power comes a greater responsibility to do good.
Extra Notes:
Where ptotal is the momentum of all the particles the object is made up of. You could also say:
This is true, even if you don't believe me. I would happily derive this for you, but I like to keep my posts to the point.
So, in order to move, you have to change your momentum (from zero to something). To do this, you need an external force (so not you nor the chair). There are two external forces, but they cancel (gravity and the force of the floor pushing up). The result is that rocking back and forth will move the chair, but the chair will move right back when you return to your initial configuration.
To demonstrate this, I created a calculation in Vpython. This calculation has two masses, one on top of the other while on a floor. There is some driving force between the two masses to simulate the rocking motion. For this first case, the top mass only oscillates horizontally (although added vertical motion would not essentially change the results when no friction is present).
Below is a graph of the horizontal position of the top and bottom mass. For this calculation, the bottom mass is four times the top mass. (I also plotted the center of mass - the black line)
In case you are still having a difficult time picturing what is happening, here is a screen shot of the setup:
Basically, the red ball is the
top mass and the yellow block is the chair.
Note in the graph above, the center of mass does not change.
Well, this is easy to fix, an external force can be added with friction. Ok - I will repeat the above calculation WITH friction. (by the way, I also measured the coefficient of kinetic friction for a rolling chair and found it to be around 0.05 - I will post this analysis later).
Notice that now, the center of mass does indeed move. However, you never really go anywhere. The average displacement is still zero. This is because the same friction force that is used to move you forward is also there when you return your head to its original position. If you want scooting to work this way, you would have to not decrease the momentum of your head (and thus you would not move back). Of course this would probably mean that your head is no longer attached to your body - generally, a bad thing.
Wait, what if you moved fast one way, and slowly back. The only way this would matter is if the static friction force is very high (I will deal with that later). For a rolling chair, this motion would still leave you with an average displacement of the center of mass at zero.
The KEY to scooting
Ok, I am ready to reveal the real key to scooting. I hope I don't offend the entertainers that rely on the mystery of scooting as part of their act - but I must do this. We can only have a free society if the secret of the scoot is understood by all.
HERE it is: Make the frictional force different for different directions. To do this, you need to know a little something about friction. Friction is actually uber-complicated, but there is a simple model that works pretty well. It says (for sliding friction):
Here N is the force the floor
pushes on the object and μ is the coefficient of
friction. So, the harder you push two surfaces
together, the more friction there is. Hey wait!
This is a rolling chair, not a sliding block.
Well, this model still workings. This is sliding
friction, with friction in the axle.
So, one way to reduce the frictional force is to reduce the force between the floor and the chair. This can be accomplished if the top mass (head) is accelerating down. This motion would cause the chair mass to accelerate up so that the center of mass stays constant. This motion can be represented with the following:
(I know this picture looks a little crazy, but let me try to explain).
The key difference is that on the left, the top object is accelerating down and to the left. The right top object is accelerating up and to the right. The vertical acceleration is very important. For the picture on the right, the top mass is accelerating down. This means that a downward force must be exerted on the top mass (by the bottom mass - chair). According to Newton's 3rd law, there must be an equal force, but opposite direction on the bottom mass. I have represented this force by the red arrow labeled "top" (sorry for including both acceleration arrows and force arrows, they are not the same thing).
Ok, so the bottom mass does not move in the vertical direction. This means that all the vertical forces must add up to zero. In this case, that is gravity, the floor and the force from the top mass. Since the top mass is pulling up, this means that the floor does not need to push as much. With a reduced force from the floor, there is a smaller friction force.
For the picture on the right, the opposite happens. The top mass is accelerating up. This causes a downward force on the bottom mass leading to increased force from the floor. Increased force from the floor means a greater frictional force.
That may be a little confusing still, let me summarize.
- Friction is related to the force the floor pushes on the bottom mass (which I sometimes call the chair)
- When the top mass (which is like the head+torso) accelerates down, this makes the floor push LESS on the bottom mass and decreases friction
- When the top mass accelerates up, the floor pushes MORE on the bottom mass and increases friction.
So, what happens if I include both horizontal AND vertical motion in my model? In this case, I created a force between the top and bottom mass that looked like this (on the top mass)
Note that the amplitude of the
force in the y direction is 2 times that in the
x, this is just because it gives better results.
(also note that the force on the bottom mass
will be the negative of this force).
I know you are waiting for the results, you have been very patient. Here is a plot of the motion of the top and bottom mass when vertical motion is also included.
I have also displayed the frictional force (purple line). When the horizontal acceleration of the top mass is the greatest negative value (greatest negative curvature) the friction is minimized. When the horizontal acceleration of the top mass has the greatest positive value, the frictional force is a maximum. This produces a steady net motion of the center of mass (as well as the position of the chair). Note that this is likely not the most optimal accelerations of the top mass to produce the most efficient movement, but it is the first thing I tried.
Real Data
Below is data from the posted video above (from a student, who I will call Mad Mursavich in case he doesn't want me to post his real name). In this graph, I have plotted the horizontal position of the head and the chair as well as the vertical acceleration of the head. This vertical acceleration was obtained by differentiating the y position of the head twice (I used Plot to do this - an awesome FREE program)
Really, I would like to plot the frictional force as I did with the numerical model, but I don't have that data. What I do have is the vertical acceleration of the head. When the head has a positive acceleration, the head will be pushing down on the chair creating a greater frictional force. When the head acceleration is negative, this will reduce the frictional force. So you see that the chair is moving forward when the head acceleration is negative (lower frictional force). Note: I scaled the acceleration data so it would fit on the same graph. Sorry, but I am not yet an expert with Plot to make this look perfect.
I guess that is it, the physics of the scoot. I know I have more to do, and I will keep you updated. I feel like the Penn and Teller of scooting now - revealing the secrets. Now that you know the secret remember that with great power comes a greater responsibility to do good.
Extra Notes:
- Analysis of the video was done with Tracker Video (another GREAT free program)
- Here is my Vpython program (its not that well organized, but oh well). Note that I wrote the data to a file so that I could create graphs using Plot. (scoot2.py)
- I reserve to right to make a mistake in my program, but the basic idea seems ok.
- Don't use these calculations for any life-saving activities.
- Don't do drugs.
