Simple Nature - Light and Matter

w / A wedge.x / Archimedes’ screwThis is an example of a simple machine, which is any mechanicalsystem that manipulates forces to do work. This particular machinereverses the direction of the motion, but doesn’t change theforce or the speed of motion.A mechanical advantage example 37The idealized pulley in figure u has negligible mass, so its kineticenergy is zero, and the kinetic energy theorem tells us that thetotal force on it is zero. We know, as in the preceding example,that the two forces pulling it to the right are equal to each other,so the force on the left must be twice as strong. This simplemachine doubles the applied force, and we refer to this ratio asa mechanical advantage (M.A.) of 2. There’s no such thing asa free lunch, however; the distance traveled by the load is cut inhalf, and there is no increase in the amount of work done.Inclined plane and wedge example 38In figure v, the force applied by the hand is equal to the one appliedto the load, but there is a mechanical advantage comparedto the force that would have been required to lift the load straightup. The distance traveled up the inclined plane is greater by afactor of 1/sin θ, so by the work theorem, the force is smaller bya factor of sin θ, and we have M.A.=1/sin θ. The wedge, w, issimilar.Archimedes’ screw example 39In one revolution, the crank travels a distance 2πb, and the waterrises by a height h. The mechanical advantage is 2πb/h.3.2.10 Force related to interaction energyIn section 2.3, we saw that there were two equivalent ways oflooking at gravity, the gravitational field and the gravitational energy.They were related by the equation dU = mg dr, so if we knewthe field, we could find the energy by integration, U = ∫ mg dr,and if we knew the energy, we could find the field by differentiation,g = (1/m) dU/ dr.The same approach can be applied to other interactions, for examplea mass on a spring. The main difference is that only in gravitationalinteractions does the strength of the interaction dependon the mass of the object, so in general, it doesn’t make sense toseparate out the factor of m as in the equation dU = mg dr. SinceF = mg is the gravitational force, we can rewrite the equation inthe more suggestive form dU = F dr. This form no longer refers togravity specifically, and can be applied much more generally. Theonly remaining detail is that I’ve been fairly cavalier about positiveand negative signs up until now. That wasn’t such a big problemfor gravitational interactions, since gravity is always attractive, butit requires more careful treatment for nongravitational forces, wherewe don’t necessarily know the direction of the force in advance, and168 Chapter 3 Conservation of Momentum