Fundamental Physics by Ramamurti Shankar at Yale_3_Multi Body System

It’s an interesting story about Newton that Professor Shankar mentioned. He published his law of motion and the calculation of Kepler’s observation many years later. The reason is that he went on to invent entire integral math to solve the problem pf treating earth as an abstract point or not.

This is related to the physics dealing with multi bodies. By introducing a fictitious value R, R can be thought as the abstract center of mass composed of multiple objects. The law can be simplified as below.

Once we figure out the center mass concept, then we do a series of deduction on Newton’s motion equations since now the center mass can be deemed as a point/object in that world.

So if F is greater than 0, meaning there is an acceleration, the two-body or multi-body system will move in an accelerated fashion.

If F is equal to 0, there are two further scenarios. 1. the original status is moving at a constant speed, so multiply the speed with the mass m*velocity or m*dx/dt = momentum, it is the definition of momentum in physics, it indicates how hard it would be to stop an object.

If F is equal to 0, the second scenario, 2. the original status is still, at rest. then what we can deduce is that the two bodies can do whatsoever movement they want, however, the center mass will have to maintain at rest as initial.

So we can solve collision bodies on frictionless ice ground problem such as

A great practical case is to observe Earth and Sun, the two can be approximated as the above two-body system. Now if as we naively think per rough observation, the sun stays still, while the earth keeps orbiting around the Sun, that means, the center mass point will have to move around. This contradicts to both scenario of F = 0. Hence, physicists infer that the sun must have been moving around also, to keep the center mass same place. Because the Sun is much more massive than the earth, the radius is so small, it’s easily neglected by simple cosmic observers.

Grasping this theory can help tackle real-life problems such as the below jumping boat experiment. Even the person can be safely jumping a 3-meter gap, under the circumstance of standing on a still boat, the nature of two-body system will cause the boat to move away from the dock, hence, if he chooses to jump right on the maximum 3-meter position, he sure will fall off to the water. Furthermore, it’s good to think that if the person successfully reached to the ground, the boat will be moving away from shore at a constant speed, which can be calculated if we measure the speed of the person upon jumping: MVboat = mVperson. It’s also good to ask once the person jumped on the ground, he stops moving because of the friction provided by the ground, i.e. an external force came into play to interrupt the two-body zero-momentum status quo.

Another practical case is to calculate the speed of rocket before and afterwards. The caveat here is that we need to be clear that the speed of the fuel (mass of delta) is v – v0, because relative to the ground, it has its original stick-to-rocket speed of v, then back-firing speed of V0, using negative to indicate the opposite direction.

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