Absolutely Small - Michael D. Fayer [8]
CAUSALITY FOR BIG OBJECTS
The importance of being able to make any object big is that it can be observed without changing it. Observing an object without changing it is intimately related to the concept of causality in classical mechanics. Causality can be defined and applied in many ways. One statement of causality is that equal causes have equal effects. This implies that the characteristics of a system are caused by earlier events according to the laws of physics. In other words, if you know in complete detail the previous history of a system, you will know its current state and how it will progress. This idea of causality led Pierre-Simon, Marquis de Laplace (1749-1827), one of the most renowned physicists and mathematicians, to declare that if the current state of the world were known with complete precision, the state of the world could be computed for any time in the future. Of course, we cannot know the state of the world with total accuracy, but for many systems, classical mechanics permits a very accurate prediction of future events based on accurate knowledge of the current state of a system. The prediction of the trajectory of a shell in precision artillery and the prediction of solar eclipses are examples of how well causality in classical mechanics works.
As a simple but very important example, consider the trajectory of a free particle, such as a rock flying through space. A free particle is an object that has no forces acting on it, that is, no air resistance, no gravity, etc. Physicists love discussing free particles because they are the simplest of all systems. However, it is necessary to point out that a free particle never really exists in nature. Even a rock in intergalactic space has weak gravity influencing it, weak light shining on it, and occasionally bumps into a hydrogen atom out there among the galaxies. Nonetheless, a free particle is useful to discuss and can almost be realized in a laboratory. So our free particle is a hypothetical true free particle despite its impossibility.
The free particle was set in motion some time ago with a momentum p, and at the time we will call zero, t = 0, it is at location x. x is the particle’s position along the horizontal axis. The trajectory of the rock is shown in Figure 2.5 beginning at t = 0. The momentum is p = mV where m is the mass of the object and V is its velocity. The mass on earth is just its normal weight. If the rock is on the moon, it has the same mass, but it would have one-sixth the weight because of the weaker pull of gravity on the moon.
A very qualitative way to think about momentum is that it is a measure of the force that an object could exert on another object if they collided. Imagine that a small boy weighing 50 pounds runs into you going 20 miles per hour. He will probably knock you down. Now imagine that a 200-pound man runs into you going 5 miles per hour. He will probably also knock you down. The small boy is light and moving fast. The man is heavy and moving slow. Both have the same momentum, 1000 lb-miles/hour. (lb is the unit for pound.) In some sense, both would have the same impact when they collide with you. Of course, this example should not be taken too literally. The boy might hit you in the legs while the man would hit you in the chest. But in a situation where these types of differences did not occur, either would have essentially the same effect when running into you.
FIGURE 2.5. A free particle in the form of a rock is shown moving along a trajectory.
Momentum is a vector because the velocity is a vector. A vector has a magnitude and a direction. The velocity is the speed and the direction. Driving north at 60 mph is not the same as driving south at 60 mph. The speed is the same, but the direction is different. The momentum has a magnitude mV and a direction because the velocity has a direction. In Figure 2.5, the motion is from left to right across the page.
At t = 0 we observe (make measurements of) the rock’s position and momentum. Once we know x and p at t = 0, we can predict the trajectory