Johnny5

How about look towards polaris? Yes no?

Power has units of joules per second, which is energy per unit time. Here is the definition of torque: Torque = T = r X F If we disregard the direction of the torque, we can focus on it's magnitude which is given by: |T| = |r||F| sin(r,F) Torque has units of energy. If you differentiate both sides with respect to time, each side will have units of power. d|T|/dt = d/dt (|r||F| sin(r,F)) So, here is what happens in a car engine... Gasoline is ignited, actually a mixture of gasoline vapor and air, regulated by the carburator, inside a chamber, by a spark which comes from the spark plug. So there is a tiny explosion which pushes the piston in a valve down, and then there is a firing sequence, if you have a four valve, there are four spark plugs. So the plugs fire in an order, i think its 1324. But regardless, as the pistons go up and down, the camshaft is turned. A DOHC is the easist engine to learn first. Dual overhead cam. So energy in the gas, is converted into rotational energy. There are places on the web where you could investigate how a real DOHC engine works, they will have diagrams and you can see what I'm talking about. But the principle is very basic, the explosions are used to make the wheel axis spin. Camshaft animation So the goal is to take the given information, and calculate torque. You are told that the maximum power delivered is 95,000 watts. One watt is a joule per second. Now, in that torque formula, you see the symbol r, you have to know what that is. I've always called it the "moment arm." I'm sure there are other terms for it. If you take a class, either in Statics or Dynamics, you will come across that formula. here is one way to think about the torque formula: You are walking towards a door, which is ajar. You want to enter the room beyond, so you have to push the door open. The door hinges are to your left, and the doorknob to your right, and the door opens inwards, that is you have to push. So where is the intelligent place to apply force? You can either use trial and error, or you can use the torque formula and deduce the answer. Here is the correct answer, regardless of your method of learning it: Do not push near the hinges, push as far away from them as you can. You want to maximize the moment arm R, so that you can apply the least force possible, to produce a given torque. Now, the hinges are located along the axis of rotation of the door. A typical door has a width of about 4 feet, so we will call that the maximum radius, at which you can apply the external force F. Now, when you push, you can either push perpendicular to the two dimensional plane which the door is in, or you can push at some other angle. So what would be the best choice for the direction of your applied force? Again, either you can use trial and error, or you can use the torque formula. Now, the letter r, and the letter F, as well as torque T, that you see in the torque formula are actually vectors, and which i will now boldface, so... t = r X F I just switched to the Greek letter tau, instead of T, because it is common practice to use that symbol to denote torque. Now, here is the definition of moment arm r, in the formula above, in the context of the door problem... The vector r, has its tail located along the axis of rotation, its head is located at the exact place of contact, at which the force is applied, and r vector is perpendicular to the axis of rotation (which we know is located where the door hinges are). (From experience, you should already realize that if you push the edge of the door towards the axis of rotation, the door will not rotate at all, no matter how hard you push) So let me take the time to explain the cross product briefly. The cross product of two vectors, is another vector, which is perpendicular to the other two. And the direction is given by the right hand rule. I should be able to find a place on the web with an illustration. Here is a brief article on "right-hand rule" at Wolfram, I strongly urge that you read it, and then come back. As far as I know, the direction is by convention. Hence this is something you must memorize. Now, I will use the diagram at Wolfram, to talk about the cross product of r with F. As you can see in the diagram, the vector u X v is perpendicular to both u, as well as v. Let that axis be the axis of rotation, and let U be the moment arm vector, so that v is the applied force vector. Thus, the direction of the torque points along the axis of rotation. Now, here is the formula for the magnitude of torque again: | t | = |r||F| sin(r,F) sin(r,F) stands for the sine of the angle between r, and F. if the angle between r,F is zero, then the torque would be zero, because sin(0)=0. Thus, there would be no "turning force" the door would not spin, and this corresponds to pushing the edge of the door directly towards the axis of spin. So clearly the door won't spin, if the angle between the moment arm r, and the applied force F is zero. Now, sine takes on its maximum value when the angle is 90 degrees. Thus, torque is maximized precisely when the angle between r and F is 90 degrees. In fact, sin(90)=1. So now back to the car engine problem. The force is being applied to a steel axis, namely the wheel axis. Let me try to find the radius of an actual wheel axis, in a car assembly.

Speaking of hyperphysics, do you know who runs that site? It is seriously well organized. I mean if i only read their whole site, I would learn a few things. Do you know anything about who designed it?

Would you please elaborate, because under certain conditions, they (reading less, ticking slower) would correspond, under others they would not. Regards PS: Perhaps this should be moved to another thread, since the original poster wanted to talk about hypersphere. You know, sometimes it is difficult to stay precisely on topic.

I am trying to stay on topic, so it will have to be in another thread. It's not hard at all. Kind regards Dr Mattson ?

Well unless i introduce coordinate systems, it's hard to know exactly what I did. Then after having done that, we can get very mathematical. I felt the more relaxed Gonzo Bonzo error was sufficient to elucidate the point, rather than introducing coordinate transformations from one frame to the other. Is that what you want to see?

Quaternions Is this correct? Did William Rowan Hamilton invent the vector cross product, by inventing quaternions?

I am going to start myself off, generate my own knowledge as it were. Here is a definition for "moment of inertia" Definition: moment of inertia So at least that is a start, if anyone here objects to this definition, please tell me why. PS, completely disregard the theory of relativity. I would like a total non relativistic treatment of the subject. The reason being, is that my toy top is not spinning at the speed of light. Now, here is something they say at the site, which I now have permanently memorized: The above is quite understandable. Every object has its own unique center of mass, which can be described in its own rest frame. This was explained in the mechanics book i was talking about, but the author did a poor job explaining it. To paraphrase... the greater the concentration of material away from the objects center of mass, the larger is its resistance to being made to spin faster. (resistance to angular acceleration) Right now I am fixated on a bicycle wheel for some reason. This site also mentions the parallel axis theorem which i dont know. I vaguely remember it, but I surely did not memorize it. As I recall, once I accept the universality of conservation of angular momentum, the rest follows. Perhaps someone will take that route. Once latex is working again, I am going to go heavy on the mathematical treatement of this. Definition: L = r X P r is the moment arm, F the applied force. All three are vectors. Ok here is a question, who came up with this formula, it is like so totally non-intuitive... I = moment of inertia = S mi (ri)2 People just don't know things like this by magic. Here is the thing which I apparently keep missing: This site is good. Ok they kind of start off with this: Angular momentum in classical mechanics That's a place to start. Now, they aren't clear where this origin is to be located though.

I want to understand the following formula: t=I a Where t denotes torque, a denotes angular acceleration, and I denotes the moment of inertia tensor. I've been doing a great deal of reading, looking for someone who understands it more clearly than any other, and I've had no luck. I really really want to understand "moment of inertia tensor" Is there anyone who has a deep understanding of it, who can shed some light. Even historical information is welcome. Like who developed the formula, when why how. I have those kinds of questions. Also, I think this thread might ultimately be quite valuable to others, who don't understand that formula either. I would like the thread to be moderated, remain on topic, and be professional. I would like it to proceed from the simple to the complex. I will not ask dumb questions, and am prepared to do some independent study. I currently have gathered several books which discuss it to varying degrees, but again... I haven't read a single person's coverage of the matter, to the point where I can say, wow that guy's good. There are some brilliant people that frequent this site, and I know this. I want to understand precession especially. The boomerang question also got me back to this question. I've never understood the moment of inertia tensor, so I can rectify that now. Umm... lets see... there was something else I wanted to say... Oh yeah... Yesterday, I was thinking about how I would proceed to teach the topic, if i already understood it, and I came up with this... I would buy a tiny toy spinning top. Then, to my pretend student, I could repeatedly perform a simple experiment. Set the top spinning on the floor, and have them watch the precession of the axis of rotation. Then, slowly I would develop a mathematical model of the toy. I could mark places on the top, where imaginary axes go. I could say, there is a frame, attached to the top, whose axes spin with it. The axes of this spinning frame pass through the points I've made on this toy top, and one of the axes of this three dimensional coordinate system is permanently the axis of rotation. So we need to come up with formulas which predict the precession of this top, which motion we can see by repeatedly performing the experiment. I might then talk about density, mass per unit volume of the top, then again perhaps not. I might touch on the integral calculus, and how it is used to "discuss" the location of the center of mass of the top. But what I really have no idea about, is how to get to "moment of inertia tensor" from just watching the thing spin. So that's what I really want to do. Thank you PS: Now here is wolfram on moment of inertia Here is a site that is beyond understandable: Rotation of rigid bodies Now in something I was recently reading, it was Euler who extended Newton's laws to rotational motion. I don't know how true that is. There are "rotational analog" formulas, to Newton's linear ones. I know the formulas, here is a link: Rotational Dynamics At the site above, they say "fixed axis" I'm not sure what that means. Here is a site that looks good: Rotational Dynamics And here is exactly what I was looking for, a side by side comparison of the formulas: Hyperphysics on Moment Of Inertia I think the first question to answer, is why doesn't the spinning top fall to the ground, it stays up in a manner of speaking. If you nudge it, it pushes back, to remain spinning. And also, there is the ice skater. She pulls her arms in, and spins faster. I know many of you know how to handle this topic. I am hoping for a clear presentation, something i cannot seem to find in my books. I recently went back to a classical mechanics book, written by umm I can't remember his name, but it doesn't matter. He glossed over things. his treatement of the subject was ultimately incoherent. I could tell he understood it on some level, yet well here I am asking for someone who knows how to teach to try to explain. I really think the presentation needs to be based upon some real experiment with a spinning top, and not be overly mathematical. however, you have to go from a spinning top, to tensors, so eventually things get abstract. I'm ok with that. but if someone were to come to me right now, and ask me why doesn't a spinning top fall to the ground, I would be unable to impress them with my answer. That's going to change.

Dr Swanson, can you explain why. You are right, under the assumption that the SRT time dilation formula is true, you are right. The frames are not identical. But how would you explain why mathematically? Your discussion would involve derivatives with respect to time of SRT formulas yes? That is what I'm interested in seeing. How you perform the derivatives. which frame do you differentiate time in. Regards

Well i have hundreds of books, graduate level mostly. In the overwhelming majority, ln is used exclusively for natural log. And in complex variables, Ln (x) is used. I'd prefer to see people use ln for natural log, Log for log base 10, and log for arbitrary logarithms. But of course i wont get what i want.

This is something I would like to see.

I love that site... Wolfram. Regards

Ok I'll check it out. I was going to do the circular version first, I feel that is a bit more difficult to understand. I myself like challenges, when things are too easy i tend to pay them no mind. I read your post about "squeeze," should i infer that is what you are doing with your attack on SRT? Doing that, in a sense, to the SRTists? If that is what you meant with that remark, it was very funny. Regards

SR has formulas in v. Differentiate them to give you formulas in a. Then investigate the meaning of the symbolic statements, and decide whether or not the meaning is inconsistent with something you already know. This is a rather complex task by the way. Regards

Not exactly in quite those words no.