Consider a force in one d with magnitude and direction given...

consider a force in one-d with magnitude and direction given by F(x)=(-4x^3)+2x. (a)ascertain the points at which the force vanishes (b)Determine the potential energy function associated with this force, whose reference value is U(0)=0. (c)For each of the points in (a), indicate whether a particle located at that point would be "stable" or "unstable." A classical potential energy function applicable to the hydrogen atom and to the solar system has the form U(r)=-(constant)/(r) where r is a radial coordinate and the constant has units of Jm (a)Glossing over the fact that the hydrogen atom and the solar system are fully three-dimensional, determine the form of the conservative force which is associated with this form of potential energy (b)Suppose that a particle subject to the conservative force(in (a)) moves radially inward from ri to rf=ri/2 (i)Under this change in position the potential energy of the particle (increases/decreases/remains the same) (ii)Assuming that no non-conservative forces act while the particle's radial position changes, the kinetic energy of the particle (increases/decreases/remains the same)

Transcript

00:01 I review here the relationship between force and potential energy.

00:06 So force is typically a vector, but if we're in one dimension, we can just write force as a scalar function.

00:15 It is related then in a one -dimensional case to the negative derivative of the potential energy with respect to the variable, whether that variable is a linear dimension x or in a centripetal sense, the radial direction r.

00:36 And that means that you can also find potential energy by integrating the force versus position.

00:50 And that theoretically gives you a change in potential energy.

00:54 So there is usually a reference point involved at the bottom limit of the integral.

01:02 And that is true for both.

01:04 Force as a function of x and force as a function of r.

01:14 One last thing in the analysis is recall that equilibrium means force is equal to zero, whether that's f of r or f of x.

01:33 And there is a way to analyze the stability of that force.

01:39 So by taking the second derivative of you, for example, with respect to x, that is minus the slope of the force graph.

01:57 And if your potential energy is concave up, you have what's called a stable equilibrium.

02:11 That's what's called stable.

02:13 And you can think about the harmonic oscillator potential energy, which goes basically, as x squared.

02:23 If the mass is sitting right at that initial x -equal zero, it will stay there in a stable manner.

02:34 And conversely, if the potential energy is, the second derivative is negative, that is a potential energy that is concave down at the equilibrium position.

02:54 And that is definition of unstable equilibrium.

03:01 So think of a ball sitting at the top of a hill versus a bottom of a hill.

03:09 A ball at the top of a hill will have a tendency to roll down, and at the bottom will have a tendency to stay put.

03:19 So here we'll take a look at a couple examples.

03:22 The first example will be a linear force in one dimension, f as a function of x, is minus 4x cubed plus 2x.

03:39 And what does that look like? you can make a graph of it, and it basically kind of looks like a, almost like a sine wave, but it's not a sine wave.

03:51 Near the origin, so one of the equilibrium positions you can certainly tell is x equals zero, but there are a couple other equilibrium positions.

04:03 So let's go ahead and find those.

04:19 So x equals 0 is of the positions, and another position set of positions occurs when x squared equals 1 half, or x is plus or minus 1 over the square root of 2.

04:40 We may find the potential energy function, and actually this is delta u, but we are going to assume a reference position of zero, so that is just simply you of x final, if you want to think about it that way, minus zero is equal to minus 4x cubed, plus 2x is the integral of a polynomial, and that gives us x to the fourth minus x squared.

05:41 And we'll replace the x final with just a regular x variable...

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