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0:00
Aditya Panjiyar
(I) A 7150-kg railroad car travels alone on a level frictionless track with a constant speed of 15.0 m/s. A 3350-kg load, initially at rest, is dropped onto the car. What will be the car's new speed?
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Keshav Singh
(II) According to a simplified model of a mammalian heart, at each pulse approximately 20 $g$ of blood is accelerated from 0.25 m/s to 0.35 m/s during a period of 0.10 s. What is the magnitude of the force exerted by the heart muscle?
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Kathleen Tatem
(I) What is the weight of a 68-kg astronaut ($a$) on Earth, ($b$) on the Moon ($g =1.7 m/s^2$) ($c$) on Mars ($g = 3.7 \,m/s^2$) ($d$) in outer space traveling with constant velocity?
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Lydia Guertin
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Yeah, in this video will derive more expressions for the electric potentials generated by different charge distributions. First, we'll consider an infinitely long charge line. This line has of the New York charge density of Lambda in units of cool apps for meter. And we'll recall that line such as this has an electric potential that is dependent on the radial distance from this line in that potential as a magnitude of Lambda over two pi her absolutely not in the radio direction. So for this point here, Director are. And as we move around the circle, we're always pointing away from the line with our electric field. So again we calculate our potential with respect to two different points, which is e d l. But of course, both e he always points in the radial direction. We take this from a to B, and so when we pull out our Constance, we simply get the integral for me to be of d. R over our and this gives us Lambda over two pi Absolutely not Times the natural log up be over a So now we have to decide where is our zero? We've always set this BB equal to zero in order to define where that is, because potential is always relative and we have to have some place where zero Well, if we set the potential FB to be zero at infinity, well, it's not possible. This term blows up and the entire potential becomes infinite. So we might instead think, Let's set be at the wire. But if we do that, then we get something fortunate Ln of zero, which is undefined. So we can't do that either. And so what we often do in a case like this, because we let B b some random distance from the wire we often call that distance are not. And so now we can rewrite this as R B of our where are can replace a as a placeholder Be lambda over two pi epsilon, not times the natural log of are not over our. So when r equals are not, we get zero potential. As we go farther and farther away, our becomes larger, this number becomes smaller. Our potential becomes more and more negative since the log oven number less than one is negative. As our becomes smaller than are not, this term here becomes larger and larger, and we get a larger and larger potential, as we would expect. Close toe A charge like this, assuming Lambda is positive. So our next example is a charge conducting sphere and will recall that in the conductor the charge will spread around such that the field on the interior of the conductor are in the interior of the conductor is always zero. And because of the spherical symmetry in this case, the charge which totals and Q will spread evenly all around the surface so as to minimize the repulsive forces between those charges. So let's consider the potential. And first we'll start at a point outside this fear. Now we know when we're outside the sphere, we can treat the entire thing as a point charge centered the center of the sphere. So for little are greater than big are or little ours just that distance we know our electric field is simply of that of a point charge to be cake over r squared in the art direction. Now we once again calculate our potential. We integrate this along some path, but since electric field is always in the radio direction, this is simply e D R. Which gives us K times Q and the integral of D r over R squared from A to B and we've already done this. We know the answer. At least four point charge where we have cake over be plus cake over a and again we set our potential to be zero at infinity and that works well. B is infinity. This term goes to zero. And so our full potential is merely that. Or as we can, right for our greater than our V of R or B. As a function of our is cake over our, we expect this to recover our equation for the potential of a point charge. But now what happens inside? Well, we know the electric field is zero. So for our less than our, we can still say B A minus. B B is the integral of the electric field times. They are from A to B, but of course the electric field changes values as we pass through the surface. Once we get to our equal capital, our angle inside the spirit, the field become zero. And so we have to split this integral and consider it first from some eight a big art, easier plus big are to be er and we know the field here in the first term is always zero because we're on the interior of the sphere. So we get zero there, and as we demonstrate above, well, that potential go to zero as B goes to infinity and now we just get cake over our, which you'll notice is also the limit here as we approach there, that's always a good sanity check. Make sure that your potentials cross boundaries are equal when you reach those boundaries from all sides. So now we can summarize by saying or potential as a function of our is equal to cake over. Our are greater than capital are and cake over capital are for our less than capital are now One more thing I want to point out, and we'll get some room to work with Here you might recall what the electric field looked like graphically where we have our electric field because of this conducting sphere and on the other axis we have our radio distance and this daughter lion is our well, we know inside sphere our electric fields zero and then we hit some maximal value here and it falls is one over r squared where this maximal value is cake over capital R squared. And so, for comparison, let's graph the potential as a function of our and again we're gonna see a point here capital or where this happen where we get a change in behavior. Now we know that as we approach, we reached some maximal value at little are equal to bigger, and we fall off as one over our It looks similar to this, except this curve over here should be more quadratic. And once we're inside, field is zero. So the potential remains constant because there's no field there to create more or less potential. And of course, this value here simply que que over a capital r. There's a good visual representation of what the field and the potential look like in all of space for this conducting show
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