Rotational Dynamics/Angular Momentum

In summary: Only in this case the picture is at just one instant and the wheel is rolling and the whole picture is moving forward. But all the little arrows are pointing either straight up or straight down. Each arrow is at right angles to the direction of motion of its point.Now here is the key part. If you draw a line from the tip of each little arrow thru the point at the center of the tire, you get a funny shape, right?The shape is called a CYLINDRICAL surface. The radius of the cylinder is the length of the arrow. The cylinder points straight up or down. And
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I'm at the end of a 1st semester college Physics class. I was handling it pretty well until we got to Rotational Dynamics/Angular Momentum. I could and will probably survive by practicing problems and recognizing the mechanical steps required to solve these certain types of problems, but I really want to acquire a fundamental understanding of the concepts involved rather than just be able to repeat the steps required.

Can anyone recommend some good, clear, easy to understand reading on these topics?

Thanks.
 
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Fundamental law of rotational motion is obtained by differentiating the definition of angular momentum L=[rxp] over time which results in dL/dt = T, where T=[rxF] is labeled as "torque". So, whatever the torque vector direction is, the angular momentum vector changes in that direction.
 
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Originally posted by discoverer02
I'm at the end of a 1st semester college Physics class. I was handling it pretty well until we got to Rotational Dynamics/Angular Momentum. I could and will probably survive by practicing problems and recognizing the mechanical steps required to solve these certain types of problems, but I really want to acquire a fundamental understanding of the concepts involved rather than just be able to repeat the steps required.

Can anyone recommend some good, clear, easy to understand reading on these topics?

Thanks.

I will be watching to see if anyone does come up with some (especially online) explanatory reading. Other people could probably use it too.
Many people are impeded by the vector "cross product".

How do you feel about cross product? Is it intuitive for you or does it put you off?

Some people are slowed down by wondering WHY a rotation in a plane should be represented by a vector which is perpendicular to the plane and whose length is an index of the rate of rotation.
and direction of rotation.

Or, when angular momentum is measured instead of simple rotation rate, the length of the vector reflects the amount of angular momentum about that axis. and direction.

Discoverer02, you haven't said where the difficulty is, for you.
Different sorts of people get stuck on different things.
I can only GUESS that you are having problems with cross product and the algebra of cross products and you are wondering what the reason for this way of representing rotation and angular momentum. But there may be other things.

It is not unusual for first year students to by frustrated with these things and actually takes some work and practice---hard work and hard practice---to get thru it. But then intuitive understanding will come. As long as the prof is not a jerk.
 
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Originally posted by marcus
Some people are slowed down by wondering WHY a rotation is a plane should be represented by a vector which is perpendicular to the plane and whose length in an index of the rate of rotation.
and direction.
Actually, rotation in a plane isn't represented by a vector orthogonal to the plane and so on... it's represented by a 2-form created from the wedge product of two 1-forms "pointing along" each axis of the plane... and it just happens that in Euclidean 3-space, such an object is is indistinguishable from a vector.

;)

- Warren
 
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I find a few different things confusing. Cross product is one of them. I don't, however, have a problem with the mechanics of the algebra. That's pretty straight-forward. The whole notion of the cross product, and just what exactly r x p physically is. isn't clear to me. Other little things like the velocity at the top of a tire being twice as great as the velocity of the center of mass and how the velocity of the tire at the bottom is temporarily zero is also a little hard to understand. Because I don't find some of it intuitive, it's difficult to keep all the concepts straight and know when and sometimes how to apply them.

So I'm looking for something that explains the topic as a whole and puts everything together, and not just a specific piece.
 
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To undertsand tire, move with it. Then in your system of reference tire center is not moving (0), the tire bottom is moving back (i.e. with -v versus you) and the tire top - forward (with +v). Now return back to non-moving system (=add v to all), and you'll get 0 at bottom, v at center and 2v at top.

(Called addition of velocities due to coordinate transformation from moving to non-moving system).

In non-moving system tire is seen as rotating around its bottom ( instant axis of rotation constantly moving forward with rire).
 
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Originally posted by Alexander
[
In non-moving system tire is seen as rotating around its bottom ( instant axis of rotation constantly moving forward with rire). [/B]

This is the part that I can't comprehend -- the notion that the tire is rotating around its bottom when it's moving forward and spinning around it's center.
 
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Originally posted by discoverer02
This is the part that I can't comprehend -- the notion that the tire is rotating around its bottom when it's moving forward and spinning around it's center.

If we are lucky someone will show up who knows a good online or other textbook chapter on rotation/ang.mom.

But until that happens...

The picture of the tire is an instantaneous one. Pretend you had a camera that could take a picture of the tire just as it rolled past you and that it would show, painted on at each point, a little arrow that is the instantaneous velocity of that point-----at the instant the shutter clicked.

The snapshot of the tire would be all covered with little arrows.

The point touching the road would have a zerolength arrow.

The point at the top of the tire would have a horizontal forwards pointing arrow twice the length of the one at the center of the hubcap

Up the center of the picture would be layer after layer of arrows getting longer the higher off the road, till reaching the top of the tire. Their length would be proportional to height.

Just like on a regular stationary turning wheel the speed of a point is proportional to how far out from the center.

None of the arrows in the snapshot are correct except for that one instant (well, the center of hubcap arrow stays the same but the rest dont)

The very next instant you would need to erase them all and draw the picture over, because a new point would be touching the ground. But the picture would look the same! It will always look,
for just that one instant, like rotation around the point touching the ground.

One can ask, well OK but what is the big deal? why should a physics teacher want me to realize this. But as long as one doesn't get off onto that and just looks at the tire it is not such a hard picture to get. Actually I like it---it helps solve something in an elegant way, doesn't it? Did this discussion help at all?
 
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Discoverer, imagine rolling hoop (ring) instead of tire (for simplicity). In co-moving reference system the hoop is simply rotating, thus each point of hoop has velocity v directed tangentionally. Now take any point on a hoop and add to this velocity same velocity vector v but directed always forward. The vector sum of the two vectors will be the same as the velocity vector of rotating around bottom point tire.
 
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Originally posted by marcus



The very next instant you would need to erase them all and draw the picture over, because a new point would be touching the ground. But the picture would look the same! It will always look,
for just that one instant, like rotation around the point touching the ground.

One can ask, well OK but what is the big deal? why should a physics teacher want me to realize this. But as long as one doesn't get off onto that and just looks at the tire it is not such a hard picture to get. Actually I like it---it helps solve something in an elegant way, doesn't it? Did this discussion help at all?

Thanks marcus. I can see the importance of the point and how conceptually what happens about this point would be considered rotation.
 
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Originally posted by discoverer02
Thanks marcus. I can see the importance of the point and how conceptually what happens about this point would be considered rotation.
My pleasure, discoverer! Sorry none of us came up with an good online tutorial on rotation-physics! I should know a link like that but unfortunately don't. Hang in there. Guess it must be final exam time. Try PF again when you have more free time, maybe we will have suggestions then.
 
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BTW do you have any specific areas of puzzlement
you want to talk about-----I don't seem to have a URL
for a general rotation-physics tutorial
 
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seems we are going in circles here.

my physics professor says " I always know when someone is going to blow the test (when they are taking it) when i see them staring at their left hand"
 

1. What is rotational dynamics?

Rotational dynamics is the branch of physics that studies the motion of objects that rotate around an axis. It involves concepts such as torque, angular velocity, and angular acceleration.

2. What is angular momentum?

Angular momentum is a measure of an object's tendency to continue rotating around an axis. It is calculated by multiplying the object's moment of inertia by its angular velocity.

3. What is the relationship between torque and angular acceleration?

Torque is the force that causes an object to rotate, while angular acceleration is the rate at which an object's angular velocity changes. The relationship between the two can be described by the equation: torque = moment of inertia x angular acceleration.

4. How does rotational dynamics differ from linear dynamics?

Rotational dynamics differs from linear dynamics in that it focuses on the motion of objects that rotate around an axis, while linear dynamics studies the motion of objects in a straight line. Additionally, rotational dynamics involves concepts such as moment of inertia and angular velocity, which are not present in linear dynamics.

5. What are some real-world applications of rotational dynamics?

Rotational dynamics has many practical applications, including the design of engines and turbines, the motion of objects in space, and the behavior of spinning objects such as tops and gyroscopes. It is also important in fields such as engineering, robotics, and sports, where understanding the rotational motion of objects is crucial.

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