Skip to main content
Loading Explore guides…
Pendulum Problem · Acedra
Home
Explore
Physics
Pendulum Problem
Sign in to save
Chat
If YouTube asks you to sign in, open the video on YouTube.
Watch on YouTube
Reload player
Transcript
131 segments
Copy
Hide video
0:00
Luca Guazzotto: Hello, and welcome to another work. Example Luca Guazzotto: for the Luca Guazzotto: Classical Mechanics course in 2,023. Please read the problem. I will fix the type of while you read it. and then we can get started. Luca Guazzotto: Okay, while you read the the problem, we also made a little drawing to get an idea of the geometry Luca Guazzotto: here is in green is the pendrum length, L. Mass. M. It's attached. Its pivot point is attached to Luca Guazzotto: circle.
0:35
Luca Guazzotto: and it moves with
0:38
Luca Guazzotto: velocity. Luca Guazzotto: angular velocity, Omega. So
0:44
Luca Guazzotto: the motion is in this direction. and the angle formed by the
0:54
Luca Guazzotto: position of the pivot and the horizontal is going to be omega times T. And this length is a. The angle between the pendulum and the vertical is theta and Luca Guazzotto: other color.
1:11
Luca Guazzotto: Of course.
1:14
Luca Guazzotto: let's just use a different color. Luca Guazzotto: gravity points. Luca Guazzotto: No.
1:32
Luca Guazzotto: Okay. Now we hopefully have everything Luca Guazzotto: equations of motion. Well, the first thing you have to wonder is, how many degrees of freedom does the system have? Luca Guazzotto: And the answer is
1:47
Luca Guazzotto: maybe not so obvious. Maybe it is. The answer is actually one theta, the angle. The motion is in the plane, so I only need to give the angle the the position
2:00
Luca Guazzotto: of the mass. M
2:05
Luca Guazzotto: as coordinates X and Y. Luca Guazzotto: But since
2:12
Luca Guazzotto: the mass is attached to an oscillating pendulum, and we know as a function of time the position of the pivot or the pendulum. In fact, once I give theta, I can determine the Luca Guazzotto: coordinates X and Y for the mass, and I'm going to
2:30
Luca Guazzotto: shoulders. do you in a second first thing. Luca Guazzotto: and some one X is going to be
2:40
Luca Guazzotto: to equal to Luca Guazzotto: a times cosine of omega T. That's the position of the the X Luca Guazzotto: of the pivot. So let me call it text, P.
2:55
Luca Guazzotto: Plus
2:58
Luca Guazzotto: b times the sine Luca Guazzotto: of theta.
3:04
Luca Guazzotto: And I'm going to call
3:09
Luca Guazzotto: X of the pendulum Luca Guazzotto: can do that.
3:15
Luca Guazzotto: Let me call it excel. and the position relative to the PO. 2. Dependent. And for y. well, I'm going to have the position of the pivot a times sine of Omega T.
3:31
Luca Guazzotto: Where it's okay minus since y points up and the pendulum is down. From that. the position of the pivot minus B Luca Guazzotto: times and observance lobby itself.
3:53
Luca Guazzotto: Sorry for that I was just following the alphabet.
3:57
The recording is called Luca Guazzotto: Times, the cosine of Theta. Luca Guazzotto: which I think I'm called. Luca Guazzotto: Why, sometimes
4:08
Luca Guazzotto: that's the complete position of in the pendulum. Then I can get
4:16
Luca Guazzotto: the velocities. This is position.
4:24
Luca Guazzotto: Then I can get the velocities. Salon me. Luca Guazzotto: I suppose. I just jumped in this promise so fascinating. But what am I going to do?
4:35
Luca Guazzotto: Well, I want to write the Lagrangian for the system that will make it very easy for me to write the equations of motion in order to write the Lagrangian and need the kinetic energy in the potential energy. Luca Guazzotto: The kinetic energy is going to be given by one half for the mass times the velocity square. So I need Luca Guazzotto: an expression for the velocity. Easiest way to do it is to use
5:02
Luca Guazzotto: Cartesian coordinates take find x of T and y of t, and they take derivatives Luca Guazzotto: of
5:12
Luca Guazzotto: x and y, and find the x and DY, Luca Guazzotto: and that will give you an expression for the velocity square, and therefore Luca Guazzotto: for the kinetic energy. As for the potential energy, since gravity points down, all I need is Luca Guazzotto: why of my months?
5:34
Luca Guazzotto: Okay? Then I found the position.
5:38
Luca Guazzotto: Then I take the derivatives with respect to time and define the velocities derivative of the cosine is minus sine. So, good man, sigh. Omega spine omega t. Luca Guazzotto: The derivatives of the sign is equal sign plus. let's see. Luca Guazzotto: do it this way. L. Times Cosine of theta times, the derivative of theta with respect to time, that's theta dot Luca Guazzotto: and y.is similarly a times omega cosine of omega t derivative of the first term
6:17
Luca Guazzotto: minus l times the derivative of the cosine, which is minus the sine, therefore, plus l. Luca Guazzotto: Then I have a theta dot again sign Luca Guazzotto: of Theta.
6:34
Luca Guazzotto: So far, I hope this was not
6:38
so difficult.
6:42
Luca Guazzotto: but now that I have the velocity.
6:47
Luca Guazzotto: I can
6:50
Luca Guazzotto: do the square. let's see, do I really want to do it, or just tell you what the answer is. Okay.
6:58
Luca Guazzotto: I don't want to do it. Let's do it, anyway.
7:04
Luca Guazzotto: X dot square plus y dot square gives me the velocity square, and I multiply by one half of M.
7:12
Luca Guazzotto: Now
7:15
Luca Guazzotto: from X dot square, I'm going to have Luca Guazzotto: a squared omega squared sine squared omega t from y dot. I'm going to have a square omega squared cosine of omega t squared.
7:28
Luca Guazzotto: So I just take the sum of the 2 and write it as a square omega squared. Luca Guazzotto: Similarly, I have l squared theta dot square cosine square plus l squared theta dot square, sine, squared
7:42
Luca Guazzotto: and square, the dot square.
7:47
Luca Guazzotto: and then I have the cross terms the annoying ones
7:52
Luca Guazzotto: have minus.
7:55
Luca Guazzotto: and a omega theta dot
7:59
Luca Guazzotto: sine omega t cosine theta
8:09
Luca Guazzotto: plus LA omega beta dot cosine of omega t sine theta.
8:22
Luca Guazzotto: and for this I will Luca Guazzotto: ask you to
8:28
Luca Guazzotto: go find it and
8:30
Luca Guazzotto: some textbook. But this is a classical sign. Luca Guazzotto: Alpha Cos beta. or cos Alpha sine Beta type of
8:46
programmatic expression. Luca Guazzotto: Long story short.
8:53
Luca Guazzotto: I'm just going to tell you what the answer is. Well, it's not the answer. There is nothing wrong about the way it's written, but we can make it more compact, and I'm going to use a square bracket here now.
9:08
Luca Guazzotto: the first 2 terms are okay, as they are. The last one is Luca Guazzotto: also okay, but don't like it too much. So let me Luca Guazzotto: modified a little bit
9:23
Luca Guazzotto: smarter. I'm missing some dos here.
9:33
This turns out to be
9:38
Luca Guazzotto: to a Luca Guazzotto: theta dot square times a times. Omega
9:46
Luca Guazzotto: fires the sign of Theta minus negative.
9:53
Luca Guazzotto: Notice that this is not the only way to write it. but that's the keynote
10:08
Luca Guazzotto: for the potential energy use. It.
10:13
Luca Guazzotto: U is equal to m times. G times y. I'm just going to set Luca Guazzotto: the potential energy equal to 0 for y equals 0. What was the expression for y, we still have it up here that's going to be equal to n times. G
10:28
times Luca Guazzotto: sine of Omega t minus L. Luca Guazzotto: Cosine of theta.
10:44
Okay.
10:51
Looks good.
10:55
Luca Guazzotto: I'm going to square it, even though this was close to 0 work.
11:10
Luca Guazzotto: And then I can write the Lagrangian as T minus u.
11:16
Luca Guazzotto: so one half of
11:21
Luca Guazzotto: a square omega square plus n squared theta dot square
11:28
Luca Guazzotto: plus 2, and theta dot and omega sign of Luca Guazzotto: theta minus omega t
11:41
Luca Guazzotto: minus. and g
11:45
Luca Guazzotto: a sine omega t minus L. Luca Guazzotto: Cosine of Theta.
12:03
Luca Guazzotto: And now we can write the equations of motion.
12:11
Luca Guazzotto: I'm going to need
12:17
Luca Guazzotto: the derivative of the Lagrangian with respect to Theta Dot, and then I'm going to take the derivative with respect to time.
12:30
Luca Guazzotto: Let's do
12:33
Luca Guazzotto: oops this derivative first.
12:36
Luca Guazzotto: while theta dot only appears inside the square bracket
12:42
Luca Guazzotto: and going to have one f of n first time does not have the dot.
12:51
Luca Guazzotto: so I'll
12:55
Luca Guazzotto: and square theta dot
12:58
Luca Guazzotto: plus 2 and 8 Luca Guazzotto: omega
13:04
Luca Guazzotto: sign of theta minus omega. T,
13:08
Luca Guazzotto: then I take the derivative with respect to time. Now, here you have to be a little careful. But let me get rid of the tools. First.
13:27
Luca Guazzotto: okay. so
13:31
Luca Guazzotto: take the time derivative of this quantity.
13:37
Luca Guazzotto: there is a total derivative, not a partial derivative. So I'm going to have the derivative respect to time of theta dot. Luca Guazzotto: which is data double dot and square
13:50
Luca Guazzotto: plus l times a times omega. These are all constant. Luca Guazzotto: Then I have the derivative of the sine of Omega t, which is the cosine sorry
14:03
Luca Guazzotto: theta minus omega t. And then I have the derivative of the argument
14:15
Luca Guazzotto: which is
14:17
Luca Guazzotto: going to be. Let's see, how do I write it? There is no confusion. Do it this way.
14:26
Luca Guazzotto: Derivative with respect to time of theta minus omega t, which is theta dot minus omega.
14:35
Luca Guazzotto: The other piece
14:38
Luca Guazzotto: is the derivative of the Lagrangian. With respect to Theta.
14:43
Luca Guazzotto: it's fairly easy to the first 2 terms
14:48
Luca Guazzotto: don't depend on theta explicitly. Luca Guazzotto: The last term does the third term does
14:58
Luca Guazzotto: tops. I lost mass here. This all multiplies and
15:05
we're gonna be in trouble.
15:08
Luca Guazzotto: Okay, so this gives me and I get rid of the 2 times l Luca Guazzotto: times theta dot times a times omega Luca Guazzotto: times the cosine of Theta, minus omega T,
15:27
Luca Guazzotto: and the other piece
15:30
Luca Guazzotto: use me.
15:35
Luca Guazzotto: Miles
15:38
Luca Guazzotto: and times. G. Times. L Luca Guazzotto: derivative of the cosine is minus the sine. Luca Guazzotto: so
15:48
Luca Guazzotto: that leaves me
15:53
Luca Guazzotto: with the correct signs.
16:04
Luca Guazzotto: Now, mmm.
16:15
Luca Guazzotto: I see that there is a mass everywhere, so I can divide everything by the mass. I cannot actually actually also divide by M.
16:25
Luca Guazzotto: Let me do that. and I find that l theta double dot
16:32
Luca Guazzotto: plus
16:35
Luca Guazzotto: a omega times theta dot minus omega Luca Guazzotto: times, the cosine of theta minus omega t. Luca Guazzotto: The the first piece is equal, so I don't have. Luca Guazzotto: What about the next one sign Luca Guazzotto: to set a dot
16:55
Luca Guazzotto: a times omega
17:00
Luca Guazzotto: times the cosine of theta minus omega t
17:07
Luca Guazzotto: minus
17:12
Luca Guazzotto: g times. That's g naught. Y Luca Guazzotto: sign Luca Guazzotto: of theta
17:24
Luca Guazzotto: next Luca Guazzotto: stuff were pretty much done. Luca Guazzotto: But if you notice
17:32
Luca Guazzotto: this term here. Luca Guazzotto: to the right time, sorry. This term here
17:39
Luca Guazzotto: is identical to this term here. and they have the same sign on opposite side of the Luca Guazzotto: equal, so I can Luca Guazzotto: through them.
17:49
And Luca Guazzotto: I can
17:54
Luca Guazzotto: this point pretty much sold for Theta double dot. If I'm so inclined and sure, why not? Theta double dot is equal to? What do I have left? I have this minus omega, I'll take that I take Luca Guazzotto: to the right hand side, so it becomes plus
18:12
Luca Guazzotto: a omega square over L
18:18
Luca Guazzotto: times cosine of theta minus omega t.
18:22
Luca Guazzotto: minus G
18:26
Luca Guazzotto: sign of thinking.
18:29
Luca Guazzotto: And that's the equation of motion for my system.
18:38
Luca Guazzotto: Now, I would like to
18:42
Luca Guazzotto: encourage you to maybe spend 10 min Luca Guazzotto: and try to do this in using Newtonian mechanics and see what you guide.
18:52
Luca Guazzotto: and I encourage you to spend only 10 min on it.
18:59
Luca Guazzotto: and then give up because it's going to be an item. But this. Luca Guazzotto: from my point of view, wasn't too bad. It's all straightforward, and Luca Guazzotto: we found an expression for theta double dot. Now solving this is going to be far from easy. but in principle. Luca Guazzotto: and but but we have the differential equation for theta double dot which we may solve numerically may also
19:30
Luca Guazzotto: find the approximation for small theta small oscillations.
19:38
Luca Guazzotto: and the way we will do for the normal pendulum. Hmm.
19:44
Luca Guazzotto: okay. So that is. problems
19:52
Luca Guazzotto: was checking my result. The world don't even look funny. This is G over. L. Luca Guazzotto: Notice that if I set Omega equal to 0. I just get this regular pendulum. which is what I should expect.
20:07
Luca Guazzotto: Okay, thank you for listening and stay tuned for the next problem. Thank you.
Here to help you learn
Sign in to keep a private conversation with your course tutor.
Ask me anything about the material…
Sign in to chat