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ation,we get the same thing as though we added before the differentiation ∑F:=F=“(Σm) Therefore the total force is the second derivative of the masses times their positions, added togethe Now the total force on all the particles is the same as the external force. Why? Although there are all kinds of forces on the particles because of the strings, the wiggling, the pullings and pushings, and the atomic forces, and who knows what and we have to add all these together, we are rescued by Newton's Third Law. Between any two particles the action and reaction are equal, so that when we add all the equations together, if any two particles have forces between them it cancels hich arise from other particles which are not included in whatever object we decide to sum over. So if Eq.(18.3)is the sum over a certain number of the particles, which together are called"the object, "then the external force on the total object is equal to the sum of all the forces on all its constituent particles Now it would be nice if we could write Eq. (18.)as the total mass times some acceleration. We can. Let us say M is the sum of all the masses, i. e, the total mass Then if we define a certain vector r to be R (184) Eq( 18.3)will be simply F= d?(MR)/dr2= M(d'R/dr2), since M is a constant. Thus we find that the external force is the total mass times the acceleration of an imaginary point whose location is R. This point is called le center of mass of the body. It is a point somewhere in the "middle "of the object, a kind of average r in which the different ri 's have weights or importances proportional to the We shall discuss this important theorem in more detail in a later chapter, and we shall therefore limit our remarks to two points: First, if the external forces are zero, if the object were floating in empty space, it might whirl, and jiggle, and twist, and do all kinds of things. But the center of mass, this artificially invented, cal culated position, somewhere in the middle, will move with a constant velocity of a box, perhaps a space ship, with people in it, and we calculate the location of to stand still if no external forces are acting on the box. Of course, the space ship may move a little in space, but that is because the people are walking back and forth inside: when one walks toward the front, the ship goes toward the back so as to keep the average position of all the masses in exactly the same place Is rocket propulsion therefore absolutely impossible because one cannot move the center of mass? No but of course we find that to propel an interesting part of the rocket, an uninteresting part must be thrown away. In other words, if we tart with a rocket at zero velocity and we spit some gas out the back end, then this little blob of gas goes one way as the rocket ship goes the other, but the center of mass is still exactly where it was before. So we simply move the part that we are interested in against the part we are not interested in. The second point concerning the center of mass, which is the reason we introduced it into our discussion at this time, is that it may be treated separately from the"internal"motions of an object, and may therefore be ignored in our discussion of rotation 18-2 Rotation of a rigid body Now let us discuss rotations. Of an ordinary object does not simply rotate, it wobbles, shakes, and bends, so to simplify matters we shall discuss the motion of a nonexistent ideal object which we call a rigid body. This means an 18-2
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