Simple Nature - Light and Matter

ordrdt = dx dy ˆx +dt dt ŷ + dzdt ẑ .All of this reasoning applies equally well to any derivative of a vector,so for instance we can take the second derivative,ora x = dv xdt ,a y = dv ydt ,a z = dv zdtdvdt = dv xdt ˆx + dv ydt ŷ + dv zdt ẑ .A counterintuitive consequence of this is that the accelerationvector does not need to be in the same direction as the motion. Thevelocity vector points in the direction of motion, but by Newton’ssecond law, a = F/m, the acceleration vector points in the same directionas the force, not the motion. This is easiest to understand ifwe take velocity vectors from two different moments in the motion,and visualize subtracting them graphically to make a ∆v vector.The direction of the ∆v vector tells us the direction of the accelerationvector as well, since the derivative dv/ dt can be approximatedas ∆v/∆t. As shown in figure y/1, a change in the magnitude ofthe velocity vector implies an acceleration that is in the direction ofmotion. A change in the direction of the velocity vector producesan acceleration perpendicular to the motion, y/2.Circular motion example 72⊲ An object moving in a circle of radius r in the x-y plane hasx = r cos ωty = r sin ωt ,andwhere ω is the number of radians traveled per second, and thepositive or negative sign indicates whether the motion is clockwiseor counterclockwise. What is its acceleration?⊲ The components of the velocity arev x = −ωr sin ωtv y = ωr cos ωt ,and for the acceleration we havea x = −ω 2 r cos ωta y = −ω 2 r sin ωt .andandThe acceleration vector has cosines and sines in the same placesas the r vector, but with minus signs in front, so it points in the oppositedirection, i.e., toward the center of the circle. By Newton’ssecond law, a=F/m, this shows that the force must be inward aswell; without this force, the object would fly off straight.z / This figure shows an intuitivejustification for the factproved mathematically in theexample, that the direction of theforce and acceleration in circularmotion is inward. The heptagon,2, is a better approximation toa circle than the triangle, 1. Tomake an infinitely good approximationto circular motion, wewould need to use an infinitelylarge number of infinitesimal taps,which would amount to a steadyinward force.Section 3.4 Motion In Three Dimensions 209