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3.) You learned in Physics 221 that all conservative forces, F, have an associated potential energy, U, and they are related by the equation: $U(x) = - \int F(x)dx$ A a.) If the Coulomb force has an associated potential energy given by the expression: $F(r) = \frac{A}{r^2}$, what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero. A b.) If the Coulomb force has an associated potential energy given by the expression: $F(r) = \frac{A}{r}$, what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero. 

          3.) You learned in Physics 221 that all conservative forces, F, have an associated potential energy, U, and they are related by the equation:
$U(x) = - \int F(x)dx$
A
a.) If the Coulomb force has an associated potential energy given by the expression: $F(r) = \frac{A}{r^2}$, what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero.
A
b.) If the Coulomb force has an associated potential energy given by the expression: $F(r) = \frac{A}{r}$, what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero.
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3.) You learned in Physics 221 that all conservative forces, F, have an associated potential energy, U, and they are related by the equation: U(x) = - ∫ F(x)dx A a.) If the Coulomb force has an associated potential energy given by the expression: F(r) = (A)/(r^2), what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero. A b.) If the Coulomb force has an associated potential energy given by the expression: F(r) = (A)/(r), what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero.

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University Physics with Modern Physics
Hugh D. Young 14th Edition
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3.) You learned in Physics 221 that all conservative forces, F, have an associated potential energy, U, and they are related by the equation: U(x) = -∫ F(x) dx a.) If the Coulomb force has an associated potential energy given by the expression: F(r) = (A)/(r^2) what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero. b.) If the Coulomb force has an associated potential energy given by the expression: F(r) = (A)/(r) what would be the corresponding expression for the force, U(r)? A is a constant. I must see your setup of your integral. Assume the constant of integration is zero.
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Transcript

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00:01 In this problem, you're given a force magnitude and your tunnel is always pointing away from some point in space.
00:10 So if this is the point you want to look at the force on an object here, that force is always away from it.
00:16 Always outward along the radio direction.
00:19 You might say, i've never seen anything like this before.
00:21 Yes, you have.
00:22 Just not outward.
00:23 You saw it inward.
00:24 Think of universal gravitation, always toward the center of the earth.
00:29 So you've seen it.
00:29 And you also knew that that force, this is not one of our squared.
00:36 That force was conservative.
00:38 This force also is conservative.
00:39 That's why they're asking you to get a potential energy function.
00:42 We could go through a lot.
00:43 We could do a whole other problem analyzing this in the same way to justify that.
00:51 That it is conservative that you can write a potential energy.
00:55 But we don't need to do that here.
00:58 So let us look at this.
01:00 Delta u the changing potential energy is defined so this is u r i'm going to go from infinity to r and that's defined as minus the integral from infinity to r f dot d r prime i have an r here i want to have a different symbol here so we don't have any confusion just a dummy index i'll be able so it's okay don't be bothered by that.
01:34 Now, dr prime vector is dr prime, r hat prime.
01:41 Just think of this as the r prime now.
01:44 So, anytime i dot a vector with a unit vector, i get the component in that direction.
01:49 So it becomes minus infinity to r, f, r prime, dr prime.
01:57 That's what i get.
01:59 Now, as i have drawn, that is a positive rp.
02:04 Prime components.
02:06 It's in the positive r hat prime direction.
02:10 So now i can put in, this is minus infinity to r, a, e minus b, r prime, d, r prime.
02:24 And now we can do the integral.
02:26 This is minus a, and the integral of e to minus b r prime is minus b, e to minus b, and our limits infinity to r.
02:38 And we can clean this up a little bit.
02:42 A over b, eta minus b r prime, infinity to r.
02:50 And i'll put in our limits.
02:52 A over b, e minus br, minus e to minus b times infinity, but that's one over an extremely large number.
03:04 Back to the infinite.
03:05 This term does not contribute.
03:07 This term is gone.
03:08 And let's not forget they tell us also that you, we're to take you at infinity to be zero.
03:12 So this is ur now.
03:17 So it becomes a over b, e minus b, r.
03:22 That's the functional form.
03:24 Now they ask us to put it right for position d.
03:28 R is equal to d.
03:29 So we would write u at d, a, e minus b, d over b...
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