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5. A bicycle wheel is mounted at the end of the physical pendulum. The bicycle wheel can be free to rotate about its axis; Or it can be tied down by & red cord,...

Question

5. A bicycle wheel is mounted at the end of the physical pendulum. The bicycle wheel can be free to rotate about its axis; Or it can be tied down by & red cord, a8 seen in the photograph below, s0 that it cannot rotate at all_If the wheel is tied down; S0 that it cannot rotate about its axis, it oscillates with some period: The pendulum is allowed to make ten oscillations while the clock runs; so the period is the final measurement on the clock divided by ten: Now suppose that the red cord t

5. A bicycle wheel is mounted at the end of the physical pendulum. The bicycle wheel can be free to rotate about its axis; Or it can be tied down by & red cord, a8 seen in the photograph below, s0 that it cannot rotate at all_ If the wheel is tied down; S0 that it cannot rotate about its axis, it oscillates with some period: The pendulum is allowed to make ten oscillations while the clock runs; so the period is the final measurement on the clock divided by ten: Now suppose that the red cord tying the wheel is released, allowing the wheel to rotate about its axis; if it wants to. The pendulum will be pulled to the side &s in the case above and again released from rest, but with the wheel free to rotate about its axis. Will the time period of oscillation be bigger or smaller than what we had before when the cord was tied? Give quantitative details Take the mass of the wheel m to be much greater than that of the pendulum; the length of the pendulum to be L and the radius of the bicycle wheel to be R



Answers

Pendulum with Varying Length. A pendulum is formed by a mass m attached to the end of a wire that is attached to the ceiling. Assume that the length $I(t)$ of the wire varies with time in some predetermined fashion. If $\theta(t)$ is the angle in radians between the pendulum and the vertical, then the motion of the pendulum is governed for small angles by the initial value problem
$$\begin{array}{l}{l^{2}(t) \theta^{\prime \prime}(t)+2 l(t) l^{\prime}(t) \theta^{\prime}(t)+g l(t) \sin (\theta(t))=0} \\ {\theta(0)=\theta_{0}, \quad \theta^{\prime}(0)=\theta_{1}}\end{array}$$
where g is the acceleration due to gravity. Assume that
$$l(t)=I_{0}+l_{1} \cos (\omega t-\phi)$$
where $l_{1}$ is much smaller than $l_{0}$ l0. (This might be a model
for a person on a swing, where the $pumping$ action
changes the distance from the center of mass of the swing to the point where the swing is attached.) To simplify the computations, take $g=1$Using the Runge-Kutta algorithm with $h=0.1$ study the motion of the pendulum when $\theta_{0}=0.05, \theta_{1}=0, \quad l_{0}=1, l_{1}=0.1$ $\omega=1,$ and $\phi=0.02 .$ In particular, does the pendulum
ever attain an angle greater in absolute value than the
initial angle $\theta_{0} ?$

Everyone. So, uh, today we're gonna be taking look at a pendulum of varying length. So this problems actually pretty nice to us. It gives us the differential equation we're going to solve, um, before we even start. So we don't have to mess around with Newton's laws or anything like that. So even though it's going to be, ah, we're just gonna plug it in. I have my room. Gay Kuta method silver up here. We're gonna be doing this numerically on the computer. It's still always is a good idea to start any differential equation problem by writing down your initial conditions. So for this, we are given that fada of zero data of zero equals points 05 And we are given that, say, the prime of zero. So the first derivative of data is just going to be zero. So the angles not changing a time equals zero. Now, this problem is, it could be a little misleading because we have two functions of tea. We have data and we have l. But where you're explicitly given else, so there's no need to solve for it. So just keep in mind that we're solving this differential equation to figure out explicitly what the function fada of tea is and you'll see in the equation I've subbed in. They give us that G equals one. So just to make it easy ever to put that there, Um right. So next we're not gonna need it right yet, But they do give us l and we are going to need to enter it into our room. Gay Kuta Method Calculator. So just to have it later, we can write l of tea equals and subbing in for Constance again. One plus 10.1 coasts of tea minus points 02 Now you'll see we also have an l prime in an l squared term. But because we're explicitly given elope those air fella really straightforward assault for But we will need to know them so just quickly doing that. So l prime of tea. So one is a constant. So that's gonna go to zero and then co sign of t minus 00.0 to the argument is linear. So we can just treat it as a regular variable is that we're t and then plug in the minus point out to you. So that's negative points one sign See minus 0.2 And then l squared I'm going to save you all the algebra you can check and make sure you got what I got But oh, squared t equals one plus point 01 co sine squared of team on this point to running out of room here worth 0.2 co sign of t minus pointed to perfect. And that's Messi. But we won't have to deal with it besides propagated. OK, so the next thing we need to do is get it into the desired form for the calculator that we have chosen. So all of them are gonna be a little different, but almost all of them. In fact, this is what the room get Kuta method is based on is we need to vector rise. So this is a second order differential equation, so you can see so again only looking at data because that's variable we're solving for you see that we have a fade, a double prime term. So that's the highest derivative term. So it's gonna be second order. It also has a state of prime term. But that doesn't matter for calculating the order. It's just whatever is highest that's present. So we know you can always split up a single second order ordinary differential equation into a system of to ordinary differential equations. If we make a substitution of variables and the way we do that is, if you always start out with, um So where variables are gonna be x one of t a next to t so x one of the we're just gonna say equals fada of tea. Very simple sort of trivial state of tea. Then we're going to make X two. I see either derivative of data. So they had a prime of tea. It will become clear why we did this in a second can. Look at this in your textbook. This is a common procedure, not even just for the room. Get cooped up for solving basically any second order differential equation. It's called victimization, so we can see pretty much right off the bat that X one prime. So the first derivative of X one of key is going to be paid a prime of tea. But that's just gonna be X two. All right, so that's one differential equation in our system underlying that and then X two Prime of tea is a little less straightforward. So you see that X to prime tea is going to be equal to date a double prime of T, which is a little less straightforward. We just still no it right off the bat like we did with X one of T. But if we scroll back up to the top, we can with just basic algebra solve for state a double prime. And in doing so, we solve for X two prime. We can plug that in, and this will become X two, and the data term will become X one. And we'll have a completely self contained, uh, linear system of differential equations. Okay, so to do that and again, I'm going to spare you guys. The algebra we get this is equal to right now in terms of l negative. Too negative, too. L prime over l times x two minus sign of X one over spreads. Okay, so these were going to be your to differential equations, and we need to solve. Luckily, we don't need to analytically solve them. That would be next to impossible, especially with the limited set of methods we've developed. So far, but luckily we have computers to do that so you can find a lot of different room Dakota calculators out there on the Internet. A lot of them you can find the Kupfer Matt Lab. That's very popular option. I like E caisson Casio calculator. All of them are gonna be a little bit different. So you just have toe, really? Get in there and find out what they're asking and what their variable stand course. So you'll see right here. They have, um, X, Y and P. Those air there variables. And basically I wrote this in terms of x one, x two and t because that's the formation your book uses. But it's completely equivalent. You just have to be careful what you enter for what? So if you look at what they're describing, we have that their axes are t they're Why is our X one and their P is our X two. So basically what this is asking is for our equation for X to make sense because this is the most interesting and you'll see they just assume that you've written in this form you've done the standard X one. Crime equals x two of T. That's just the regular victimization of the second order defeat you. So our time starts at zero, can't have negative time and we just define the starting 0.0 are, um, Fada or why of time zero was point of five. And are they? The prime of zero is zero. See, these initial conditions we wrote coming back into play. And so then the only other curveball They throw you a little bit here and you'll see this in a lot of different calculators. So it's very good to know is we are given in our problem were given that h equals point one. So we go here and we see we have a slopper xn and we have a slot for partition. And so basically accent is asking you over what interval do you want to conduct this approximation? Because obviously it's a computer so they can't to an infinite interval. I would take forever. So we're just gonna put five. You can put whatever you want, you can play around with it. I encourage you to do so. But five works for our purposes here, and partition is the number of steps you would like it to do within that range. So five divided by 50 because a ranges from 0 to 5 divided by 50 equals 500.1. So that is the step length that thes conditions will give you. And you see if I could put 100 than our h would be half of what it was. So 0.5 But the problem specifically asked for H equals 0.1. So that's perfect for now. And you can press execute, you can see already did it so you can get all the data you need for this problem from this chart. Uh, there's no need to graph it, but it's always illuminating, and I've taken the liberty to graph it over on my longer pro. So here we have the graph of angle business time for variable length pendulum, and the question just asks you to analyze it so you can say a lot of things. You can say it's sandy soil, which we would expect from a pendulum, So that means we probably did something correct. You can calculate its period. You can look at its frequency, its wavelength all that would be great to put. But the thing is really getting out of the thing I'm gonna cover is that it explicitly asks you, Does this pendulum reached a greater magnitude of angle than it started at? So does he reach an angle of above 0.5 or doesn't reach an angle of below negative point of five We can see from this graph just in the 1st 2 equals five seconds that yes, it does. It goes below negative 50.5 by a fair margin, more than any margin of error, we would have add around, say, three t equals 3.2, and you can, if you expended out. I did. Up to 10 seconds can do more than that. You'll see that this does repeat itself, so it periodically reaches an angle of magnitude greater than it started. And that's the crust of the question again, include as many details as you can at all. Be great. I'm sure your teacher would love that. Um, but for all intents and purposes, we have solved this problem. So I hope you guys learn something. Have a nice day

For this problem. We have a torch in pendulum that as a metal disc with a wire running through it, um, the wires mounted to vertical clamps and pulled talked. And we have two graphs that give the magnitude of the tour and, um needed to rotate the disk about the center as well as, um, the resulting oscillations in terms of the angle and from these graphs where you're able to get that Got tau equals 4.0 times 10 to the negative. Third Newton's meter new meters. Um, data is 0.2 rads and time. A position s is 0.4 seconds are, um And we're being asked to figure out what is the rotational inertia of the disk about the center as well as what's the maximum angular speed of the disc To figure out our inertia? We're going to be you thing. The equation period equals to pie square root of inertia divided by Tor Shin, which means we need to figure out our Tor Shin. And in order to get to Orson, we're going to use the equation. Torque equals tor Shin times fade up. We have our data, and we have our torque so we can rearrange this to solve for Tor Shin, Where to Orson equals torque divided by data. So we'll have 4.0 times 10 to the negative Third, divided by our theater, which is zero point to rad. So we end up with our Tor Shin equals it's a value of zero point 02 Okay, so now we also need to figure out what our period is. And if we look at the graph here, we see that t s is the length of one full period. So in reality, our period is going to be the 0.40 seconds. So now we have our portion. We have our period, and we can go ahead and rearrange this equation to solve for inertia. And we'll get inertia equals tor shin times the period squared divided by four pi squared. Then when we plug in our values, we get that inertia equals 0.2 times 0.4 squared, divided by four pi squared, giving us a value of 8.11 times 10 to the negative fifth kilograms, times meter squared. Right now we're gonna find the maximum, um, angular speed and we're gonna use the equation. Maximum angular speed equals Seita Max times the square root of Tor Shin divided by inertial. Um, And again, our graphs here are gonna come in handy because our maximum data where our highest points here on our on this graph is point to rads. That does give us our our data, Max, for here, So plugging in the values that we know we have Omega Max is going to be equal to our theatre max, which is 0.2 times the square root of our Tor Shin, which is 0.2 divided by our inertia 8.11 times 10 to the negative fifth, which gives us a value of three points 14 rads per second.

Yes, You may have a pendulum consisting off a court, then l and a mess am attached to it. If this pendulum is allowed to swing and is then interrupted at its narrowest point, Bye. Hey, the pendulum would transcribe this circle in its motion. How do we know what the minimum possible speed at the bottom off its motion for the union, it would be to be able to pass through the entire circle. Well, let's look at what the conservation of energy tells us. Firstly, at the bottom of the circle, this is the initial point of your skills. We know you. This is the potential energy, and it is zero. The bottom of the circle is taken to be at the height zero In the kinetic energy, Kate is equal to heart. I am the i squid and this is what we're trying to find the I the velocity, as depending them begins to make circular trajectories at the top. Oh, final reference position We know you is equal to em. G bitch! And H in this case, is just how minus do you? So at this point, the top off circle, What is the kinetic energy At this point, Kate is equal to the house. M v squared? Yes, The final velocity. Well, we know that this is centripetal motion. So Connecticut energies him G times. This is each of the fact centripetal acceleration G is just the script. Do you mind it? And are in the radius of a circle is just our dynasty. That's hot. I want you now by nice No, the conservation of energy tells us that the energy you saw has to be the mechanical energy. Boston So 1/2 in your ice cream. Is it too n g to tell my STP thus Ah ha n g throughout my mystique. So some off there kinetic energy and potential in it before But the bottom of circle has to give us to some off the potential in Canada. Garrity's at the top. So if we do a bit of rearranging, we find the I to be the screams. Five things. Repetition, acceleration G to Vegas now dynasty. Okay, so the minimum velocity or the minimum speed off the pendulum At its lowest point, the I is sprayed with five g l minus steep. But what is the minimum angle, dita? Well, the minimum angle detail. Well, tell us the angle at which the pendulum needs to be the least so that the engine can pasta in the entire something again. We can use the conservation of energy. So the initial potential energy that's the final plus the initial kinetic energy Sorry physical to the final potential energy speak Final Connecticut. So pendulum starts at the height H, which is not known. It has zero kinetic energy. It stops from risk, its final potential energy zero has reached the bottom of the meat, and at that point, and that energy is 1/2 I am. And this way if the velocity that we found you, so that's just find G into out by a sneak the velocity is the square root off that. So he's quit. He's just everything under the square root. But what is H Well, with a little bit of economic street, you can see that H actually help 21 minus cost and this year's hot him into five g. How minus Well, now this looks not more promising. So you mean rearranges, Father, We see one minus course. Heat up is equal to 5/2 one minus. Yeah. No. And lastly, with a bit more mathematics, we see that this gives Peter as the arc costs both five D. Do you find it? My two l minus me. So this is the minimal angle, which the pendulum has to be released in order that it makes a sufficient velocity to complete the circle.

Okay, let's get Turk. So s so far, so far happier. I you have learned to solve the second order linear, different occasion. However, in this case, we have nonlinear differential occasion because of peace. Sign terms. So what's your first? Just tap. You have legalized thiss differential. Kishan. Right? So if you look at the hint the interest. As for a small values of Sarah, we can use to linear approximation. And so his science era is almost the same. It is Stella. So we can transform this differential equation right now. Uh, look, att Now let's move on. Lookinto, eh? Find the occasional motion of a pendulum with lengths. A one meter, Which means he said l is one. And then ah, Farah is initially pointed to radium. That means Sarah, zero is point two and then Ah, initial angular velocity is one. Right? So we have thes second order different, alina, different education. Right? So if you have ah, linear different occasion, what is your first step? Your first stubby is so the characters peopling amir. So this represents the characters to pull in. We're right. So in this case, it's a conductor. So esseker plus G you're here and she's a constant Ichi's thie Accreditation. Acceleration. Right. So if you served these characters Polina, Mira, you get too complex. Ask complex delusions. Right? So if you have a complex solutions, then what is your solution? What is your real solution? What's your solution? Off the solution to different occasion. The solution has to form or decent, right? It consists of Lisa Linear linear combination, off sign and consign. Right. And here a and B are just constant. So what is your next channel? Way? Have to find these A and B. How are using these two initial conditions? Right. So when he questions zero the Asia and Angle Wass longitude or Indian and the initial angular velocity US one. Right. So by using these two conditions, we can sew way confined to constant A and B. So if we plug zero two key, then it becomes hero. So your piece going to two and then if you differentiate it and then plug zero, then this term will vanish. So you get scare ittle g times A, of course, one which means a course one over Skerritt of G. Yeah, right. So he's easier. Finer answer to a I hear it. It is two point two, right? So let's move on. Be what is the maximum angle from the vertical? So we have to find maximum destruction. Great. And then there can be two ways. Where is it used? D'Oh Ah, extreme value deuterium. So, by finding the solution of the situation right to remember this one Yeah. By finding the solution ofthis occasion, you can find the local extreme a right and then by using using the second derivative test you can test, you can determine whether there is a local maximum or local minimum. Then you can find the maximum easily and then the second way is use former I'm not in the first way because it is not easy to compute. Thiss you know these contents So I'm going to use the formula. Dan, what's a foreigner? So when you have a function about thiss form, then you can easily and find the maximum. It is scary out ofthe spirit of thirteen. How did I do that? It is. It's karaoke east number, close piece number, and then take us in spirit. Right? So in general, if you have a constant in me, then you're mixed Mom will be scare it'LL ace care plus be scared, right? Yeah. Bye. And then by using their formal er, we can compute the maximum very easily. So scary toe one over G because I cured you. Yes, going to choose So it is point it away for right It is approximately By using calculator you can come into this So far so good, right? It's not that hard. And then that's when we want see first appearance off the pendulum So there is the time to complete back and one bank and forces tree again we can use for murder In this function we can easily find the period it iss to pry Oberst Cherie, How can I find this? So in general, if you have ah many times X in the trigger Elementary in functions break Then you are a period or B two Pi Siraj bhai, The absolute value Omi So in this case, our only that is scared of she. So we have thiss period, right? Nothing difficult So let's move on so d when will the pendulum first to be vertical? So if you look at this picture, this line is for ical So the problem asked you to find the time when this object pass through this point. So we're going to find the time. Oh, yeah. And at this point, he angle becomes you. So we have to find the time and she satisfies desiccation, set out key questions. Zero right? Then we can send this occasion Great. I just put Tio I just used to Fuller for his send off answerable t And then, uh since I want to pushed key collect tea in Oran term I divided by we'LL sign off Scared of g times t to collect tea in one term. Then we get this second occasion right here. What is this? Sign over a co sign. This is Hon. Gentle spirit of g times T right. Great. Then get in. And then here We have to think at this time a tasty she should be. How's chip? But if you find the solution on this one this one Look at this white course tenant's ex. If you remember by using Arc tangent, you confined Ah, point this point, right, well for him. But this is your never in point two times spirituality. Then you can easily find this point, however, the point is never Never Then we need a party in number so they have to line this point. So the difference between these two points is it's actually pie. So by heading high, we can get the exact solution two we'LL hear the sun becomes this so security g times ta Eucharist Our kind gentle negative points to times square to G class pie And by using calculator we can compute titties approximately. Then you get to point five pate. Am I right? Yes, this's right. Now this is an approximation. Approximately solution. So approximately tour Ah, you're going to five eight times later. But pendulum will be first. Well, first be vertical right now Let's move on. E, what's the angle of those to you? And the pendulum is for Cole. So at this point, we have to find the velocity, the angle realistic. Great. So to find the angle velocity, we have to differentiate Sarah and then pull out. Ah, just the exact time because you're kind. Yeah. So to plug this number Dan here, scourge of G times on scare I mean the tingle. I mean the tea we already find it isn't founded his soul, so to point five. So by plugging this number, we can compute it easily, then by using calculator peace ponies. Good point eight three. I mean, eight four. Right. So Ah, we can find an angle of others. See here. Notice that it is. The absolute value of this one is smaller than one, which is the initial angular velocity. Why is just happening? One? Because this is an approximately approximately solution. I mean, when you if you remember in the physics physics classes Teo, the potential energy applause, physics toe, Canary energy should be equal. So when your palate and pendulum will be the lowest, it should be fest. It shouldn't be fastest. Uh, since you used Teo approximate and more than not exact model. We have this research well, so thank you.


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5 answers
Solve the IVPdx/=. [16. 0dt. 16. 0]x(0)=[-8 -4]
Solve the IVPdx/=. [16. 0dt. 16. 0]x(0)=[-8 -4]...
5 answers
Eid #e equation @ tAe Tnacve Lne taane qivep &kveALAe ~uen Y-Value U=_XE-X When XZO
Eid #e equation @ tAe Tnacve Lne taane qivep &kveALAe ~uen Y-Value U=_XE-X When XZO...

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