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(II) $(a)$ Suppose we have three masses, $m_{1}, m_{2},$ and $m_{3},$ thatinitially are infinitely far apart from each other. Show that the work needed to bring them to the positions shown in Fig. 39 is$$W=-G\left(\frac{m_{1} m_{2}}{r_{12}}+\frac{m_{1} m_{3}}{r_{13}}+\frac{m_{2} m_{3}}{r_{23}}\right)$$(b) Can we say that this formula also gives the potential energyof the system, or the potential energy of one or two of the objects? $(c)$ Is $W$ equal tothe binding energy of the system-that is, equal to the energy required to separate the components by an infinite distance? Explain.

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A) $W=-G\left(\frac{m_{1} m_{2}}{r_{12}}+\frac{m_{1} m_{3}}{r_{13}}+\frac{m_{2} m_{3}}{r_{23}}\right)$B) Gravitational potential energy at some position ris defined to be equal tothe work done by an external force to bring the two masses from infinity tothat position. Therefore, we can say that $U=W_{e x t}$C) $|W|=G\left(\frac{m_{1} m_{2}}{r_{12}}+\frac{m_{1} m_{3}}{r_{13}}+\frac{m_{2} m_{3}}{r_{23}}\right)$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

University of Michigan - Ann Arbor

Hope College

University of Winnipeg

Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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Alright, The work required to bring these masses into position is actually just the sum, but off the work required to bring each of these masses together. So w one work required to bring mass one from infinity to ah to that point is just zero. Um, but for two w to work required to bring mass to into position, it is it is the potential energy due to the attraction between mass one and mass tune. So that's GM on him to over our want to. And for me you have the gravitational potential energy between mass lips between masses one and three g m. More than two over our 13 as well as the gravitational potential energy of masses two and three. So that's GM to our 23 gm to him three over our duty. And so therefore, the total work done is just some of all of that. And you get negative, uh, is common times g m on them, too, over our 12 are one too. Um, plus she and want to eat em on m three over our 13 plus gm two and three over our 23 and that's that some. That was party. Uh, part beef. Ah. Potential entered. This will be so. This will be the potential energy off the whole system. Okay. Potential energy does not exist for single objects of isolation. It is for entire systems that that's what it is here. Arms and part seats. The, um the binding energy will just be gross. Finding energy orb. You will just be the magnitude of the work done for a total gravity protection origin system. So it's just ju times and one and two over our 12 plus em one. And come on in three over our 13 plus and two and three over our 23 and that's it.

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