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(II) Show that Eq. 16 for gravitational potential energy reduces to Eq. $2, \Delta U=m g\left(y_{2}-y_{1}\right),$ for objects near the surface of the Earth.$$\Delta U=U_{2}-U_{1}=-\frac{G m M_{\mathrm{E}}}{r_{2}}+\frac{G m M_{\mathrm{E}}}{r_{1}}$$$$\Delta U=U_{2}-U_{1}=-W_{\mathrm{G}}=m g\left(y_{2}-y_{1}\right)$$

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$m g\left(y_{2}-y_{1}\right)$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

University of Michigan - Ann Arbor

University of Washington

Simon Fraser University

McMaster University

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.

01:03

Show that the gravitationa…

02:48

Gravitational Potential En…

02:20

Gravitational potential en…

01:19

The equation for gravi…

02:54

We normally consider the g…

02:30

Prove that the total energ…

03:47

If the mass of the earth i…

so the gravitational potential and she don't you that we start with is just, um you two mind if you want, So that's negative. G m the mass of the earth times small mass over our two. Uh, a certain height above the earth is also equivalent to y two. And, um, you subtract from that ah gmp mm over our one for another point above the earth and lower point from our two on notice that it's a negative and the negatives of is the cell. So So they cancel out and you get a positive and you can take GM time. Small am common. Um, and so what you get is GMP I'm smaller, uh, times one over r one. Because that's the positive term. Minus one over r two. That's the negative term you caused all to you. Ah, and so this is equal to hear me. Tons. Um, times are two minus R one over r. One times are too um, break the smallest playing but sides top and bottom. And so Ah, you have g times and you times m over. Our one times are too. Times are two minus are one. It's our two minutes are One is basically why to minus y one. So we already have one part of this. I've already want a show. As for the other part had let's remember that for near the surface of the earth, uh, heights are, uh most heights are most wise are much, much less than the radius of the earth. So are one is approximately equal to the radius of the earth. And same goes for our two, approximately equal to race of the earth. And so what you get therefore, is that all one times are too is approximately radius of the Earth square to a very good degree for near the earth's surface. Um, so when you get ISS, so when you get out of that is delta use CI times and you tons small m over babies of the earth squared times why? To minus y one great. And then you you make that smaller sometimes g m e over race of the earth square times why to minus y want, um and so remember that Newton's law Newton's law says that the force of gravity is negative G m you times m, uh, over ari squared and so this part. GMP, uh, this implies rather, um that the negative g m v over R b squared is actually just the gravitational acceleration acceleration due to gravity. Because, remember, at physical toe A And so for the earth a is G acceleration, and so that is equal to negative. G m e um Ah, Reconcile eyes equal. Negative. Doug is just equal to g m e over Ari squared. Great. Um, who will be a negative sign there on. So you get em times ci All this is she Times y two minus 11 and that's it.

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