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(II) A sphere of radius $r_{1}$ has a concentric spherical cavity of radius $r_{2}$ (Fig. 40$)$ . Assume this spherical shell of thickness $r_{1}-r_{2}$ is uniform and has a total mass $M .$ Show that the gravitational potential energy of a mass $m$ ata distance $r$ from the center of the shell$\left(r>r_{1}\right)$ is given by$$U=-\frac{G m M}{r}$$

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$-G m M / r$

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 Winnipeg

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.

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a little bit of algebra here. This problem, but of the first step, is to calculate the density of the show. Okay, Density of shell that extends from or want or two on DSO density, if you'll remember, is just, um, mass over volume. So density of the shell rows of H for shell eyes, math of the show are his mass m Mrs Bigham over volume just 4/3 pi r cubed. And are, uh, this k and r Cubed in this case will be Are one cubed minus are too cute because, uh, we want right. We want the density of this stuff here. Ah, not ah, anything inside or outside. So that's why we're taking the difference between the volume of a bigger circle and the volume over our big, biggest fear and volume of the smallest here. Okay, so that's Dempsey of the show. And so then will we have to do is find ah, mass of the outer shell and the, uh, inner self o r. Let's say mass m massive are one rate, so that would be the density of the shell that would remain same times. The volume of, um are one of the our lunch Ah, of the r one sphere. So that's for pod 4/3 pi R. One cubed. Therefore, you have, um, put in identity of the shell rose show. Just em over 4/3 pi are one Keep to mind if our two cubed times 4/3 pi are one cubed And so the 4/3 pies cancel giving us uh, m r one cubed over our one cubed tomatoes are too cute. Okay, um and then density off the inner shell. So that's a march to our massively in a sham or two is ro shell again times 4/3 pi r to cube this time. And so this will be most the same way m over 4/3 pi r one cube. But it's our two cubed times had 4/3 pi Uh, it's a cubed over there, but 4/3 pi are too cute. And so that gives you em. Times are too cubed over our one cubed minus r t cubed. All right, so you've got those two masses had and then you're set. All you have to do is find you shop technology of the shell, which is Ah, you are one minus you are to grab Tish potential energy at Ah, the outer shell minus gravitated potential when she had the inner shell. So this would be, um, from the formula. Negative. She, uh, upstate of hci. Times smaller, sometimes big m over are, um and so negative G m over times, eh, Marwan over art. And then that would be, um, lips. Let's Ah, look, att. Right, So that would be negative. Two. That would be negative. G m g small m times m r two over our Ah, and so you can see that negative GM over this distance are could be taken this comment. So you have negative g m for our times. M r. One minus m r two. Okay. And so will you get him some art movements. I'm somewhere too. And so replant does in from perform when you get you shell equals negative. G m are negative. G time. Small m They could say over our times m r one cute. Over are one cubed minus are too cute minus m r. Two cubed over are one cute minus R two cubed And so again we can take em outside Begin. And so we have negative g m m over our times are one cute mind if our two cubed, uh, both divided by our one cube murders are two cubed. Uh, so so this really turns out to be our one cubed minus R two cubed over our one cubed minus r two cubed which isjust one. Therefore we have that you've show equals negative G m uh, time small. I'm over our and we are done.

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