3. The Plummer model is a simple (but not very accurate)...

3. The Plummer model is a simple (but not very accurate) description of the mass distribu-\ntion of a spherical star cluster or galaxy. It has two parameters: total mass $M_0$ and \"scale\nradius\" $a$. As a function of the distance $r$ from the center of the galaxy, the mass desnity is:\n\n$ \rho(r) = \frac{3}{4\pi} \frac{a^2 M_0}{(a^2 + r^2)^{5/2}} $.\n(1)\n(a) Find the enclosed mass profile $M(r)$ as a function of radius $r$. (Hint: you'll need this\nintegral: $\int 3a^2r^2(a^2 + r^2)^{-5/2}dr = r^3(a^2 + r^2)^{-3/2}$.)\n(b) Using your result from part (a), find the rotation curve $v(r)$; that is, the velocity of a\n\"test particle\" moving in a circular orbit at radius $r$. Note that by Newton's \"iron sphere\ntheorem\", a particle at radius $r$ feels no force due to mass outside that radius in spherically\nsymmetric potentials.\n(c) Using any software you find convenient, plot the rotation curve for a Plummer model\nwith mass $M_0 = 10^{12} M_\odot$ and scale radius $a = 30 \text{ kpc}$. Compare your plot to the rotation\ncurve of the Milky Way; is it a good match?

Transcript

00:02 All right, looks like you're getting ready to do a lab and where you're going to compare what's going to happen as an object that rolls down the ramp versus what we used to do in this course where we would slide something down the ramp or use a cart where we weren't including its rotational energy.

00:26 So the first question is based on the idea of what is moment of inertia.

00:33 So right here i have the two equations for finding moment of inertia, one for the disc, one for a solid sphere.

00:40 Now a disk and a cylinder end up with the same.

00:44 So even though in your lab you're going to be doing a cylinder, the disc and the cylinder are out of the same, because it doesn't actually matter what the length of the disc is, the thickness of the disc.

00:59 Oops, i mean that.

01:02 So we can just say that in your lab, the disc in the cylinder are going to have the same moment of inertia, one -half mr squared.

01:19 Now, a spurious moment of inertia is two -fifths of mr squared, so it's less by a little bit.

01:27 Now, why is that? well, the definition of moment of inertia, i wrote it out here in terms of why one's more than the other.

01:34 Basically what you're doing is you're comparing the proportion of particles and how far away they are from the center of rotation.

01:42 So we have a center of rotation here in the center spot.

01:49 How many of those particles in proportion are far, are farther away from the center than the sphere? well, the sphere has this center spot surrounded, right? it's not only going away from it, it's also coming in and out of the page because it's a three -dimensional circle rather than a two -dimensional one.

02:14 So there's a lot more of its particles closer to the center in proportion than it is for the disc.

02:21 That is what the moment of inertia is all about.

02:24 It's how many part in proportion, how many particles are sitting around there, not the total number of particles, but the proportion.

02:32 Of them.

02:34 And so it doesn't matter what radius you have.

02:36 It doesn't matter what the mass is.

02:38 The moment of inertia on a sphere is always less than the moment of inertia of the disk.

02:43 So if we had a race between a cylinder and a sphere, no matter how big this sphere is, it will win.

02:51 It could be a small one.

02:53 It could be a large one.

02:55 As long as it's a solid sphere, it will win over a cylinder.

02:59 It always will.

03:03 All right.

03:03 So number two, you were supposed to to find the moment of inertia for your particular item.

03:09 I assume that it was 100 grams.

03:12 We need to change that to kilograms.

03:15 And then it had a radius of four centimeters.

03:18 We need to change that in meters because all this needs to be in the amount of energy.

03:24 So we can't use non -standard units here.

03:28 So meters and kilograms is what we need for this.

03:33 So that would be how you'd solve that.

03:35 Now, number three, is where i think you were having some issues.

03:38 I didn't have access to the equations that your instructor is actually referring to.

03:45 So i'm going to just assume that he's following the laws of conservation of energy here.

03:51 And he's having you compare what would happen if you don't.

03:57 What's your prediction when you're not factoring in the rolling kinetic energy? and what is the prediction when you are factoring it in.

04:08 So here's what happens when you don't factor it in.

04:12 You're saying that all of the energy that it had to start with is going to go into moving it down the ramp, straight down the ramp.

04:23 This is true if it's sliding down the ramp.

04:25 It's not, though, it's rolling down the ramp.

04:27 So that's going to, this prediction is going to be off.

04:32 This one over here, though, does factor it in, and it says that some of the kinetic energy is going to be utilizing or it's going to be moving it down the ramp.

04:42 But some of the kinetic energy is going to be used to roll it down the ramp.

04:47 What's the difference? to get it to roll requires some energy.

04:53 And that's going to have to come from the original potential energy.

04:57 So something that rolls down the ramp cannot beat something that slides down the ramp with no friction because some of its energy is lost to getting it to roll and as a have as much for the kinetic energy to go down the ramps and it'll always lose a race to something that was sliding.

05:17 Now, it's just a roll requires friction.

05:20 So it's really an irrelevant comparison here.

05:24 This is going to be inaccurate and the one on the right is going to be the correct prediction.

05:30 So i'm going to show you and i'm going to make my algebra here.

05:34 In blue so that you can see what i'm talking about.

05:39 Now, i'm not real sure what equations he wanted you, or he or she wanted you to utilize, but i'm pretty sure it's the conservation energy.

05:52 So, first thing i'm going to do is i'm going to solve for velocity on the left when we're predicting without the rotational.

06:01 Well, mass doesn't matter.

06:03 We should have learned that in a previous unit, the mass going down the ramp has no bearing on how fast it is going at the end or how long it takes it to get to the bottom.

06:15 So this is going to come down to gh is equal to one half v squared.

06:22 I'm going to rearrange this to solve for v.

06:25 So this is going to be 2gh square root.

06:30 I'll tell us how fast it's going at the bottom.

06:33 Now, what the instructor wants us to do is predict what is going to be the time.

06:40 So i need to do one more kinematic equation here to figure out, well, what's the time going to be? there's two ways i could do...