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(1I) $(a)$ Show that the total mechanical energy of a satellite mass $m$ orbiting at a distance $r$ from the center of the Earth (mass $M_{E} )$ is$$E=-\frac{1}{2} \frac{G m M_{\mathrm{E}}}{r}$$if $U=0$ at $r=\infty .(b)$ Show that although friction causesthe value of $E$ to decrease slowly, kinetic energy must actu-ally increase if the orbit remains a circle.

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A) $-\frac{1}{2} \frac{G M_{E} m}{r}$B) $-\frac{1}{2} \frac{\mathrm{GM}_{\mathrm{E} \mathrm{m}}}{\mathrm{r}}$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Hope College

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.

03:01

(II) $(a)$ Show that the t…

03:06

A satellite of mass $m$ is…

03:03

(I) For a satellite of mas…

00:46

Using the technique shown …

01:29

Two satellites of earth $\…

01:03

06:45

Neglecting Earth's ro…

06:44

Prove that the total energ…

01:44

Show that the total energy…

02:27

all right. The condition that you equal zero at our equals. Infinity. Ah, all that that says, is that you know, potential engines. Negative. Two times lots of the earth time. Small mass over our, um that's what we want now is kinetic energy and Connecticut And you. It's found by creating centripetal force absolutely to gravitational force. Uh, f g And so But that means that centripetal force and Reece squared over r is he called to gravitational for its g times? Bigham. I am Mammy told them over r squared. OK, And so one of the artist cancel. And if we most find both sides, I won't have What we have is 1/2 m v squared equals 1/2 g m times m over our and so the left hand side is obviously kinetic. Energy is calling K on the right hand side. As you can see, it's just a potential inju Time's a factor of negative 1/2. Right, So this is negative. K equals negative 1/2 you. This is the premise on so and so therefore to this is partner. Therefore, uh, total energy Ukuleles, kinetic plus potential is just cleared against negative 1/2. So it's ah o R one happening of 1/2 us are its native 1/2 you, plus you. And so that's 1/2 you and what is 1/2 you 1/2 you is 1/2 she and times m over are and this and negative there. And we haven't. That's your energy. Okay. Ah, for part B. Um, what happens when the energy decreases due to friction himself? When your energy goes down, energy goes down. What we have is from this equation. Ah, your radius will also go down. This is because of because energy is negative, so it becomes more negative. So the radius goes down. But also, kinetic energy is inversely proportional to radius as we a cz we saw here when half of b squared is inversely proportional to our, uh so kid, be, um uh are so radius has to go up our radio as radius goes down, Uh, kinetic energy goes up make. And so you have ceased to computing things. So when we want it, what will we can you can show here is that as energy goes down so as you goes down Bye one. George, what happens? You have K plus you equals. And so he is just negative as just 1/2 of U. S. O. You would go down, tell a friend that you would go down by two jewels, which means that then you have kinetic energy. It's left so kinetic energy would have to go up by one jewel. And that's how you can make sense of this energy. Total energy goes down. Your potential ghost out the kinetic would have to go up in order for the orbit to remain a circle. That's it.

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