It is nearly at all times the final matter within the first semester of introductory physics—angular momentum. Finest for final, or one thing? I’ve used this idea to explain every part from fidget spinners to standing double back flips to the motion of strange interstellar asteroids.
However actually, what the heck is angular momentum?
Let me begin with the next state of affairs. Think about that there are two balls in area linked by a spring. Why are there two balls in area? I do not know—simply use your creativeness.
Not solely are these balls linked by a spring, however the crimson 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—similar to this.
Sure, this can be a numerical calculation. If you need to check out the code and play with it your self (and you must), here it is. If you need all the small print about the best way to make one thing like this, check out this post on the three body problem.
Once we see stuff like these rotating spring-balls, we take into consideration what’s conserved—what does not change. Momentum is an effective instance of a conserved amount. We are able to outline momentum as:
Let me simply make a plot of the whole 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. Here is what I get.
In that plot, the crimson curve is the x-momentum of the crimson (heavier) ball and the blue curve is for the yellow ball (yellow does not present up within the graph very properly). The black line is the whole momentum. Discover that as one object will increase in momentum, the opposite object decreases. Momentum is conserved. You possibly can do the identical factor within the y-direction or the z-direction, however I believe you get the concept.
What about power? I can calculate two forms of power for this technique consisting of the balls and the spring. There’s kinetic power and there’s a spring potential power:
The kinetic power is dependent upon the mass (m) and velocity (v) of the objects the place the potential power is expounded to the stiffness of the spring (okay) and the stretch (s). Now I can plot the whole power of this technique. Observe that power is a scalar amount, so I haven’t got to plot only one element of it.
The black curve is once more the whole power. Discover that it’s fixed. Power can be conserved.
However is there one other conserved amount that could possibly be calculated? Is the angular velocity conserved? Clearly it’s not. Because the balls come nearer collectively, they appear to spin sooner. 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’d simply have the identical worth and never add as much as a relentless.
OK, however there is one thing else that may be calculated that can maybe be conserved. You guessed it: It is referred to as the angular momentum. The angular momentum of a single particle is dependent upon 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’s 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 end result (as a result of you possibly can’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 a substitute 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’ll say that as A cross B).
You may 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 would be able to at all times 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 at all times perpendicular to each A and B—thus that is at all times a three-dimensional downside. Oh, it’s also possible to rotate the vectors through the use of the right-click or ctrl-click and drag.
However now I can calculate (and plot) the whole angular momentum of this ball-spring system. Truly, I am unable to plot the angular momentum since that is a vector. As an alternative I’ll plot the z-component of the angular momentum. Additionally, I would like to select a degree about which to calculate the angular momentum. I’ll use the middle of mass for the ball-spring system.
There are some essential issues to note on this plot. First, each the balls have fixed z-component of angular momentum so in fact the whole angular momentum can be fixed. Second, the z-component of angular momentum is unfavourable. This implies the angular momentum vector is pointing in a path that may seem like into the display (out of your view).
So it seems that this amount referred to as angular momentum is certainly conserved. If you need, you possibly can examine that the angular momentum can be conserved within the x and y-directions (however it’s).
However wait! you say. Possibly angular momentum is just conserved as a result of I’m calculating it with respect to the middle of mass for the ball-spring system. OK, tremendous. Let’s transfer this level to some place else such that the momentum vectors would be the similar, however now the r-vectors for the 2 balls can be one thing completely different. Here is what I get for the z-component of angular momentum.
Now you possibly can see that the z-component for the 2 balls each individually change, however the whole angular momentum is fixed. So angular momentum continues to be conserved. Ultimately, 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 version here.)
Once more, angular momentum is conserved. Because the motor disk begins to spin a technique, the remainder of the platform spins the opposite approach such that the whole angular momentum is fixed (and nil on this case). For a state of affairs like this, it will be fairly darn troublesome to mannequin this example with simply forces and momentum. Oh, you can certainly do it—however you would need to contemplate each the platform and the disk as many, many small lots every with completely different momentum vectors and place vectors. It could be just about unattainable to elucidate with that methodology. Nevertheless, through the use of angular momentum for these inflexible objects, it is not such a foul physics downside.
Ultimately, angular momentum is one more factor that we are able to calculate—and it seems to be helpful in fairly various conditions. If yow will discover another amount that’s conserved in numerous conditions, you’ll most likely be well-known. You may also identify the amount after your self if that makes you cheerful.