Angular Momentum Intuition

## Question:

To whom it may concern,

I am having great difficulty with gaining an intuitive understanding of the principle of conservation of angular momentum. What I do understand is the conseravtion of linear momentum. I think I understand why linear momentum is conserved, and I believe that it is the foundation of all Newton's laws, and therefore the basis of classical physics. I can't seem to understand the conservation of angular momentum in terms of Newton's laws, however. I mean I can exploit the equations, sure, no problem, but I don't understand why they work. The major barrier in my understanding comes from this little gedanken...

Suppose I have a little robot attached to a string, and the robot is spinning around in a circle at a constant angular velocity. Let the robot then climb toward the axis of rotation so that no external torque is acting on the system. This would therefore require the angular momentum to be conserved. This is analagous to a figure skater pulling their arms in and spinning faster, but we are only dealing with an approximate point mass in this case. My problem is then this... according to the conservation of angular momentum, the the tangential velocity must increase as the radius of rotation is lessened, but how can the tagential velocity increase if there is no force acting perpendicular to the radius of rotation? An intuitive explaination would be greatly appreciated. Please don't revert to moments of inertia or energy arguments. I would like to understand this in the following sort of way...

It is given that the point mass is rotating, and so its tangential velocity is changing direction and not in magnitude. Now by moving the point mass inward, a force is exerted on the string and therefore on the axel, and since the axel is attached to the earth it can be regarded as an infinite mass. This means that the inward force must somehow cause the robot to increase in tangential velocity. The way I see it, though, is that the tangential speed would have to be conserved, and only the angular velocity would be increased. But this doesn't work in reality because according to the blasted equations, the angular velocity increases with the square of the radius. Anyway, thanks for bearing with me. Unfortunately, I think I have a severe learning disability that won't allow me to get accept something until I understand it intuitively.

Thanks a whole lot in advance.

## Answer:

This may be a less than satisfactory response but it is the best I can do at the moment. Allow me to philosophize briefly before going to the details of your question. The learning disability you mentioned is one of the most common obstacles to education. I submit that it is not so much a learning disability as an attitude problem. You were not born with that intuition that now contains linear momentum, you developed that intuition by study and observation. Intuitive understanding is a learned skill.

Now when a new concept comes along your mind's first attempt at absorbing it is to try to fit it into the existing mental framework called intuition, which you have constructed. If that fails there are two paths open to you. You may try to break the new concept down into bits that fit the old framework or you may expand the framework to hold the new concept. The latter approach requires by far the greater effort but it never fails. The former approach is less mentally demanding but sometimes it just doesn't work. Successful students are those who cheerfully go through the effort of expanding their intuition even if the payback is not immediate or even obvious. The great scholars are those who after having gone to that effort, will cheerfully throw a concept away if it turns out to that there is a better one that covers the same territory.

Now regarding angular momentum... The conservation of linear momentum carries with it an implicit assumption that in the observers reference frame the particle path is a straight line and that changes in position are measured with respect to a point on that line. For an object in circular motion, or if we stand somewhere off the line to measure the position of a particle in linear motion, there is no period of time short enough that the velocity does not change direction during that interval, violating this implicit assumption. Therefore we should not be surprised to find that linear momentum, particle mass times tangential velocity, is not conserved in operations on a particle following a curved path.

There is a more general law, the conservation of angular momentum, that contains conservation of linear momentum as a special case. The more general conservation law is that the quantity rXp is conserved in the absence of an external net torque on a particle, where r is the vector from a reference point to the particle and p is the product of the instantaneous velocity vector times the mass. This law of nature is not derivable through any amount of twisting and turning we might undertake using the laws of linear motion since those laws are below it in the hierarchy of generality. If you are familiar with all mammals you can perhaps draw a skunk but if all you know is skunks you are at a loss to draw all mammals. Same sort of principle.

When your little robot pulls itself closer to the axis of rotation, it is the quantity rXp that is conserved, not just p (or v) itself. If this is contrary to your intuition, then your intuition needs to be repaired.

This information is brought to you by M. Casco Associates, a company dedicated to helping humankind reach the stars through understanding how the universe works. My name is James D. Jones. If I can be of more help, please let me know.

JDJ