Simple Nature - Light and Matter

As in the proof on page 130, the first sum is simply the total kineticenergy in the original frame of reference, and the last sum is a constant,which has no effect on the validity of the conservation law.The middle sum can be rewritten as2n∑m j v j · u = 2 u ·j=1= 2 u ·n∑m j v jj=1n∑p j ,so the only way energy can be conserved for all values of u is if thevector sum of the momenta is conserved as well.j=13.4.6 Gradients and line integrals (optional)This subsection introduces a little bit of vector calculus. It canbe omitted without loss of continuity, but the techniques will beneeded in our study of electricity and magnetism, and it may behelpful to be exposed to them in easy-to-visualize mechanical contextsbefore applying them to invisible electrical and magnetic phenomena.In physics we often deal with fields of force, meaning situationswhere the force on an object depends on its position. For instance,figure ag could represent a map of the trade winds affecting a sailingship, or a chart of the gravitational forces experienced by a spaceprobe entering a double-star system. An object moving under theinfluence of this force will not necessarily be moving in the samedirection as the force at every moment. The sailing ship can tackagainst the wind, due to the force from the water on the keel. Thespace probe, if it entered from the top of the diagram at high speed,would start to curve around to the right, but its inertia would carryit forward, and it wouldn’t instantly swerve to match the directionof the gravitational force. For convenience, we’ve defined the gravitationalfield, g, as the force per unit mass, but that trick only leadsto a simplification because the gravitational force on an object isproportional to its mass. Since this subsection is meant to applyto any kind of force, we’ll discuss everything in terms of the actualforce vector, F, in units of newtons.If an object moves through the field of force along some curvedpath from point r 1 to point r 2 , the force will do a certain amountof work on it. To calculate this work, we can break the path upinto infinitesimally short segments, find the work done along eachsegment, and add them all up. For an object traveling along a nicestraight x axis, we use the symbol dx to indicate the length of anyinfinitesimally short segment. In three dimensions, moving along acurve, each segment is a tiny vector dr = ˆx dx + ŷ dy + ẑ dz. Thework theorem can be expressed as a dot product, so the work donealong a segment is F · dr. We want to integrate this, but we don’tag / An object moves through afield of force.Section 3.4 Motion In Three Dimensions 215