Physics/trajectory in a complex object
QUESTION: Hello, Dr. Stephen O. Nelson,
I would like to know the type of trajectory expected in the following situation:
1) There is an equilateral triangle formation of three particles connected by a string;
2) The center of mass is in the centre of the triangle as the three particles have the same mass;
3) The three particles are moving parallel to each other at v1, so the center of mass should also be travelling in the same parallel trajectory.
4) There is no inertia or any other forces to affect the motion.
So, what changes to the trajectory would be expected if...
1.1) Each particle moved at a different speed?
P1 = v1 (1 metre per second)
P2 = v2 (2 metres per second)
P3 = v3 (3 metres per second)
Attempt at an answer: the slowest particle would have a dragging effect on the others, making them rotate so as to be able to sustain their velocity (like a tri-helix). It might affect the center of mass's trajectory so that it would behave like a cilindrical spiral although maintaining its general straight trajectory.
1.2) Adding to 1.1, the particle's mass also changed?
Attempt at an answer: an increase in the mass of any one particle would increase its velocity, and a decrease would have the opposite effect, thus changing the appearance of the helix.
1.3) Adding to 1.1, how would the nature of the bonds affect the whole structure?
- if they are fixed (not elastic), the hellix will be stable (which is the approach I've taken in all of my above suppositions);
- if they are elastic, then the the hellix will have an unstable structure and its centre of mass would be always changing, allowing for the possibility of random looping (while losing its general straight trajectory);
- if one of the bonds breaks, the whole structure would stabilize in a more linear construct behaving like a double-pendulum.
As a literature and language teacher, it's been over 20 years since I've dealt with maths and physics, but I'm trying to catch up (first with Math, which is fundamental in Physics), so I will not be able to understand very complex formulas. Still, I am capable of following some degree of complexity if the symbols are explained so I can tailor my self-teaching efforts to better understanding this.
Also, where can I find (or be able to plot) a similar hellix/trajectory that fits my problem above?
Thank you for your patience and any help/guidance you may give me,
ANSWER: Well, this is a tangled web of a question. Let me make a list:
*Situation part 4) -- There has to be inertia, that's fundamental in getting particles with mass to keep moving in a straight line. I assume you mean external forces like drag and gravity.
*1.1 -- The particles are connected by strings, so it may or may not be possible for them to move at different velocities at all. For example, if their motion is along the direction of string connecting two of the particles then the one in front can't suddenly be moving faster or the string would have to break. A perfectly strong string would result effectively in an elastic collision between those two particles. How that would result in movement would be dependent on the exact directions of their velocities relative to their positions. Without a diagram it is impossible to answer this badly-posed scenario at all aside from to say that the group of them as a whole would end up moving in the same direction at 2m/s and probably in a jumbled mess.
*1.2 This doesn't specify the velocity of the mass that has been added, and cutting mass off of a particle wouldn't really have any effect. Again, not specific enough.
*1.3 This question is so general that an answer shouldn't even be attempted.
These questions are badly posed, and the scenario ridiculously complex. Where on Earth did you find such a bad problem?
---------- FOLLOW-UP ----------
QUESTION: Yes, the exposition shows that I know very little of physics indeed.
I did not find the problem - it came up from a discussion after watching a youtube video of the double pendulum (someone said something to the effect "what if" and then it grew from there):
I also thought about something which is closer to home first: if you hurl an object it rotates instead of just moving all in a straight line. I assumed this supposed particle-triangle of mine would also be doing the same thing, that is, one of its particle-angles would be ahead of the others and hence the particles would be rotating around their centre of mass.
I apologise for not presenting a diagram, but I fear I wouldn't be able to make it much better to explain this.
Ahh, yes...you probably want to limit the complexity of such systems when trying to understand physics at first. Larger systems are just built out of the smaller fundamental parts...I can tell you from experience that the mathematics of dealing with many-body systems quickly gets out of control, as does the complexity of their behavior.
So hurling a rotating object does create changing velocities for the particles that make it up with respect to the ground, as does a rolling object like a tire (the point in contact with the ground is not moving at all, the point directly above is moving at twice the speed of the wheel, for example). This is sustained by the internal forces connecting the rotating body. But the body itself will move in a straight line as a whole. I'm not sure if this helps clarify the situation, because I'm not longer sure what your question is...