Gravitational field a. Find a potential function for the gravitational field 𝐅=-G

Gravitational field a. Find a potential function for...

Gravitational field \begin{equation}\begin{array}{c}{\text { a. Find a potential function for the gravitational field }} \\ {\mathbf{F}=-G m M \frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}} \\ {(G, m, \text { and } M \text { are constants })}\\{\text { b. Let } P_{1} \text { and } P_{2} \text { be points at distance } s_{1} \text { and } s_{2} \text { from the origin. }} \\ {\text { Show that the work done by the gravitational field in part (a) }} \\ {\text { in moving a particle from } P_{1} \text { to } P_{2} \text { is }}\end{array}\end{equation} \begin{equation}\operatorname{GmM}\left(\frac{1}{s_{2}}-\frac{1}{s_{1}}\right).\end{equation}

Gravitational field
\begin{equation}\begin{array}{c}{\text { a. Find a potential function for the gravitational field }} \\ {\mathbf{F}=-G m M \frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}} \\ {(G, m, \text { and } M \text { are constants })}\\{\text { b. Let } P_{1} \text { and } P_{2} \text { be points at distance } s_{1} \text { and } s_{2} \text { from the origin. }} \\ {\text { Show that the work done by the gravitational field in part (a) }} \\ {\text { in moving a particle from } P_{1} \text { to } P_{2} \text { is }}\end{array}\end{equation}
\begin{equation}\operatorname{GmM}\left(\frac{1}{s_{2}}-\frac{1}{s_{1}}\right).\end{equation}

Key Concepts

Inverse-Square Law

The inverse-square law explains that the magnitude of the gravitational force decreases proportionally to the square of the distance from the mass causing the field. This law is fundamental in deriving the mathematical form of the gravitational field and its corresponding potential function, and it characterizes many other natural phenomena that depend on distance in a similar manner.

Gravitational Potential Energy

Gravitational potential energy represents the energy stored in an object as a result of its position within a gravitational field. It is closely related to the potential function of the gravitational field and is defined in such a way that the work done by gravity when moving an object between two points is equal to the difference in gravitational potential energy. This concept is central in analyzing energy conservation in gravitational interactions.

Line Integrals and Work in Conservative Fields

In a conservative field, the work done by a force moving a particle from one point to another can be computed as a line integral of the force along the path. However, due to the field's conservative nature, this line integral is path-independent and equals the difference in the potential function between the endpoints. This connection simplifies the calculation of work and is a key concept in both vector calculus and physics.

Conservative Vector Fields

A conservative vector field is one in which the work done moving along any path between two points depends only on the positions of the endpoints, not on the path taken. In such fields, there exists a scalar potential function whose gradient is equal to the vector field. This property simplifies the calculation of work, as it reduces to evaluating the difference in the potential function between the endpoints.

Potential Function

A potential function is a scalar function whose gradient yields the original vector field. In physics, identifying a potential function for a force field, such as a gravitational field, allows for the direct calculation of work done by the force, since the work is just the difference in the potential function at the endpoints. The potential function is defined up to an arbitrary constant, reflecting the fact that only differences in potential have physical significance.

Transcript

00:01 Hello, this is problem 30a, the last one, and it's also a little bit like physics.

00:06 First, part a asks us to find a potential function for that gravitational of view of f.

00:12 G, little m, big m are constant, so i'm going to let the whole term negative g time a little m time big m as a constant c.

00:21 So your big f now is just cx over that's whole thing, bottom of there.

00:31 Z square 2 power 3 half, i plus cy and same thing.

00:43 J.

00:44 I'm going to be lazy a little bit in this last problem, so bear with me.

00:49 Now, it's asked to find a potential function.

00:52 It's ideal to check that whether this f is exact, but i'm sure that it is exact, otherwise it won't ask us to find a potential function.

00:59 So i'm just going to go ahead and find the potential function.

01:03 With respect to x, this is ecx over x square to power three half.

01:12 So if you integrate this, you will get that is going to be something with a power five half in the bottom, and that's going to be negative 3yz, 10, c, the entire thing, everything's going to be in the bottom, but power 5 have right now, plus g that half just y, z in it.

01:42 Well, the reason for that, well, you know how to integrate.

01:47 Let's say this is just basically the integrate of here's an x.

01:57 This is d actually is d of x square plus y square plus z square, right? in term of x because that's over one half here and then that term is xxx y square plus y square plus c squared two power three half all right so you can do that um now let's go ahead and find a g so it take partial f with respect to y and this when you do partial with respect to y oh, my apology, power one half.

02:46 All right, like you do the integral.

02:48 I don't know why do you keep doing it.

02:50 I get the power three half, put one half here, right? because when you do integrate, it's negative three half, power negative three half, adding one more, so that's power negative one half, so that should be that.

03:05 Okay, all right, back to our problem here with respect to why, what do we get, when we do that with respect.

03:12 To y, where, oops, i don't know why do i get this negative 3 y, z.

03:26 Give me one moment.

03:28 When you do that, it just actually is negative c, right? negative c here.

03:38 There's a c in the front.

03:40 There's a c tab that...

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