It is virtually all the time the final matter within the first semester of introductory physics—angular momentum. Greatest for final, or one thing? I’ve used this idea to explain all the things from fidget spinners to standing double again flips to the motion of unusual interstellar asteroids.
However actually, what the heck is angular momentum?
Let me begin with the next scenario. Think about that there are two balls in house related by a spring. Why are there two balls in house? I do not know—simply use your creativeness.
Not solely are these balls related by a spring, however the pink ball has a mass that’s thrice the mass of the yellow ball—only for enjoyable. Now the 2 balls are pushed such that they transfer round one another—identical to this.
Sure, this can be a numerical calculation. If you would like to try the code and play with it your self (and it is best to), right here it’s. If you would like all the main points about make one thing like this, check out this publish on the three physique downside.
Once we see stuff like these rotating spring-balls, we take into consideration what’s conserved—what does not change. Momentum is an efficient instance of a conserved amount. We will outline momentum as:
Let me simply make a plot of the entire momentum as a operate of time for this spring-ball system. Since momentum is a vector, I should plot one element of the momentum—only for enjoyable, I’ll select the x-coordinate. This is what I get.
In that plot, the pink curve is the x-momentum of the pink (heavier) ball and the blue curve is for the yellow ball (yellow does not present up within the graph very effectively). The black line is the entire momentum. Discover that as one object will increase in momentum, the opposite object decreases. Momentum is conserved. You would do the identical factor within the y-direction or the z-direction, however I believe you get the concept.
What about vitality? I can calculate two kinds of vitality for this technique consisting of the balls and the spring. There’s kinetic vitality and there’s a spring potential vitality:
The kinetic vitality is determined by the mass (m) and velocity (v) of the objects the place the potential vitality is expounded to the stiffness of the spring (ok) and the stretch (s). Now I can plot the entire vitality of this technique. Word that vitality is a scalar amount, so I haven’t got to plot only one element of it.
The black curve is once more the entire vitality. Discover that it’s fixed. Vitality can also be conserved.
However is there one other conserved amount that could possibly be calculated? Is the angular velocity conserved? Clearly it isn’t. Because the balls come nearer collectively, they appear to spin quicker. How a few fast examine, utilizing a plot of the angular velocity as a operate of time.
Nope: Clearly, this isn’t conserved. I may plot the angular velocity of every ball—however they might simply have the identical worth and never add as much as a continuing.
OK, however there is one thing else that may be calculated that may maybe be conserved. You guessed it: It is known as the angular momentum. The angular momentum of a single particle is determined by each the momentum of that particle and its vector location from some level. The angular momentum could be calculated as:
Though this looks like a easy expression, there may be a lot to go over. First, the L vector represents the angular momentum—sure, it is a vector. Second, the r vector is a distance vector from some level to the thing and at last the p vector represents the momentum (product of mass and velocity). However what about that “X”? That’s the cross product operator. The cross product is an operation between two vectors that produces a vector consequence (as a result of you’ll be able to’t use scalar multiplication between two vectors).
I do not need to go right into a bunch of maths relating to the cross product, so as an alternative I’ll simply present it to you. Here’s a fast python program exhibiting two vectors (A and B) in addition to A x B (you’d say that as A cross B).
You possibly can click on and drag the yellow A vector round and see what occurs to the resultant of A x B. Additionally, do not forget that you could all the time take a look at the code by clicking the “pencil” icon after which click on the “play” to run it. Discover that A X B is all the time perpendicular to each A and B—thus that is all the time a three-dimensional downside. Oh, you can even rotate the vectors by utilizing the right-click or ctrl-click and drag.
However now I can calculate (and plot) the entire angular momentum of this ball-spring system. Really, I can not plot the angular momentum since that is a vector. As a substitute I’ll plot the z-component of the angular momentum. Additionally, I would like to select some extent about which to calculate the angular momentum. I’ll use the middle of mass for the ball-spring system.
There are some vital issues to note on this plot. First, each the balls have fixed z-component of angular momentum so in fact the entire angular momentum can also be fixed. Second, the z-component of angular momentum is unfavourable. This implies the angular momentum vector is pointing in a course that may seem like into the display screen (out of your view).
So it seems that this amount known as angular momentum is certainly conserved. If you would like, you’ll be able to examine that the angular momentum can also be conserved within the x and y-directions (however it’s).
However wait! you say. Perhaps angular momentum is simply conserved as a result of I’m calculating it with respect to the middle of mass for the ball-spring system. OK, high quality. Let’s transfer this level to some other place such that the momentum vectors would be the similar, however now the r-vectors for the 2 balls will likely be one thing totally different. This is what I get for the z-component of angular momentum.
Now you’ll be able to see that the z-component for the 2 balls each individually change, however the complete angular momentum is fixed. So angular momentum continues to be conserved. In the long run, angular momentum is one thing that’s conserved for conditions that don’t have any exterior torque like these spring balls. However why can we even want angular momentum? On this case, we actually do not want it. It’s fairly easy to mannequin the movement of the objects simply utilizing the momentum precept and forces (which is how I made the python mannequin you see).
However what about one thing else? Check out this fast experiment. There’s a rotating platform with one other disk connected to a motor.
What occurs with the motor-disk begins to spin? Watch. (There is a YouTube model right here.)
Once more, angular momentum is conserved. Because the motor disk begins to spin a technique, the remainder of the platform spins the opposite method such that the entire angular momentum is fixed (and nil on this case). For a scenario like this, it will be fairly darn troublesome to mannequin this case with simply forces and momentum. Oh, you could possibly certainly do it—however you would need to think about each the platform and the disk as many, many small plenty every with totally different momentum vectors and place vectors. It could be just about unattainable to elucidate with that technique. Nonetheless, by utilizing angular momentum for these inflexible objects, it is not such a foul physics downside.
In the long run, angular momentum is yet one more factor that we are able to calculate—and it seems to be helpful in fairly a lot of conditions. If you could find another amount that’s conserved in several conditions, you’ll most likely be well-known. It’s also possible to identify the amount after your self if that makes you content.