AP PHYSICS 1 AND 2 › UNIT 2, FORCE AND TRANSLATIONAL DYNAMICS › TOPIC 2.1
Systems and Center of Mass
Before a single force is drawn, one decision has to be made, and it is almost never written down: what exactly are we talking about? This topic is that decision, and the one point that is allowed to stand in for whatever you chose.
2.1.A Describe a system and its interactions, and decide when a group of objects can be treated as one object.
2.1.B Locate the center of mass of a system from its parts, and model the whole system as a single object sitting there.
1. The first line you draw is not a force
Unit 1 asked what the motion was. Unit 2 asks what is causing it, and the very first move is one that almost nobody writes down: deciding what it refers to. A cart, or the cart and the block on top of it. One climber, or the climber and the rope and the pack. That choice is called the system, and everything after it depends on which one you made.
Draw the boundary as a dashed loop. Forces that reach across it from the outside world are external, and they are the ones that change how the system moves. Forces between two things inside the loop are internal. They still exist, they can be enormous, and they contribute nothing at all to the motion of the system as a whole, because they always turn up in pairs that point opposite ways.
This is why a system can be treated as a single object at all. If what happens between the parts does not matter for the question you are asking, you may forget the parts and keep the total mass. A car is a hundred thousand pieces rubbing and burning and turning, and on a free-body diagram it is a dot with four arrows.
2. Two systems, one situation
Two blocks on a smooth floor, joined by a light rope. A hand pulls the right-hand one. Ask for the acceleration and the fastest route is to put both blocks inside the boundary: the rope forces become internal, they cancel, and one line of algebra gives the answer. Ask instead for the tension in the rope and that route tells you nothing, because you deliberately hid the rope. So you draw a new boundary around one block, and the rope becomes external, and it is the only horizontal arrow left.
Nothing about the physical situation changed. You asked a different question and drew a different loop, which is the whole skill.
Choose which blocks you are calling the system. The dashed line is the boundary, gold arrows are forces that cross it, and the faint pair is the one that does not. Only horizontal forces are drawn: the weights and the normal forces balance and would add nothing here. Watch the acceleration readout while you switch boundaries.
3. The center of mass
Treating a system as one object raises an obvious question: one object located where? The answer is the center of mass, and it is a weighted average of the positions, with each position counted as many times as it has kilograms.
xcm = (m1x1 + m2x2 + …) / (m1 + m2 + …)and the same formula again for y
Two things follow immediately. If every mass is the same, the weighting does nothing and the center of mass is the plain average of the positions. And if the object has a line of symmetry with equal mass either side of it, the center of mass is on that line, which means a uniform meter stick, a uniform disc and a uniform sphere need no arithmetic at all.
Drag the discs. Click one without moving it to change its mass. The cross is the center of mass, and the two sums above it are what put it there. Try to arrange the discs so that the cross lands on none of them, which takes about three seconds, and then ask what is actually at that point.
4. Why that point earns its name
The center of mass is not merely a convenient average. It is the one point in a complicated system that obeys a simple law. Throw a wrench across the room and every part of it does something different: the handle swings, the head loops, no single piece follows a tidy path. The center of mass draws a clean parabola, exactly as though the whole wrench were a single particle with a single force on it.
That is the promise Unit 2 is built on. When the rest of this unit says that a net force produces an acceleration, the acceleration it means is the acceleration of the center of mass. A diver can tuck, spin and open, and the center of mass never notices: the only external force is gravity, so its path was fixed the moment she left the board.
5. Three ways this goes wrong
Averaging the positions and forgetting the masses. It gives the right answer when the masses happen to be equal, which is often enough in textbook problems to hide the error for weeks. Write the masses in every time, even when they are the same.
Expecting the center of mass to be inside the material. For a ring it is the hole. For a boomerang it is the air in the notch. There is nothing there, and there does not have to be: the center of mass is a location, not a piece of the object.
Changing the system halfway through a problem. Starting with both blocks, then quietly using the tension as though it were external, produces an equation that is not wrong so much as about nothing. Name the system at the top of the page and keep it.
For this course the arithmetic stays small: five particles or fewer, laid out on a line or in a plane, or an object symmetric enough that you can point at the answer without calculating.
Check yourself
1. Two blocks joined by a rope are pulled across a smooth floor. A student says the rope tension is one of the forces that accelerates the pair. What is wrong with that, and when would the tension be an external force?
With both blocks in the system the rope pulls forward on one block and backward on the other by equal amounts, so it contributes nothing to the total. It becomes external the moment you redraw the boundary around a single block, and then it is the only horizontal force acting on that block.
2. Masses of 1 kg, 2 kg and 3 kg sit at x = 0, 2 m and 4 m. Where is the center of mass, and why is the plain average of the three positions the wrong method even though it is close?
(1·0 + 2·2 + 3·4) / 6 = 16/6 = 2.67 m. The plain average of the positions is 2 m, which counts the light mass as heavily as the heavy one. The correct answer is pulled toward the 3 kg block, which is what the weighting is for.
3. A uniform ring lies flat on a table. Where is its center of mass, and what does that tell you about whether the center of mass has to be made of anything?
At the geometric center, in the hole, by symmetry: every bit of the ring is matched by a bit directly opposite. There is no material there at all. The center of mass is a position computed from a distribution, not a part of the object, and nothing about the definition requires it to sit on matter.
4. A gymnast leaves the floor, tucks, rotates, and opens again before landing. Ignoring air resistance, describe the path of her center of mass and say why the tuck makes no difference to it.
A parabola, fixed at the instant she left the floor. Tucking rearranges her body using internal forces, and internal forces cannot change the motion of the system as a whole. The only external force in flight is gravity, so the center of mass keeps the parabola it was given no matter what the arms and legs do.
5. A student is asked for the acceleration of a two-block system and writes an equation containing both the applied force and the rope tension, then cannot solve it. Diagnose the error in one sentence, and give two different correct routes to the answer.
The equation mixes two systems: the applied force is external to the pair, the tension is not, and no single boundary makes both external. Route one: take both blocks, use only the applied force, divide by the total mass. Route two: write a separate equation for each block, with the tension appearing in both and opposite in sign, then add them and watch the tension disappear. The two routes agree, which is a useful check rather than a coincidence.
Topic 2.2, Forces and Free-Body Diagrams. The system is chosen and it has a location. The next step is to draw the arrows that cross the boundary, one straight arrow per force, and to learn which arrows students draw that are not there at all.