Instantaneous axis of rotation

[Biker]

I studied statics but I thought I can figure out the dynamics part.

In a rectangular shape that is tipping, Usually we take the center of mass as an axis of rotation however the center of mass is accelerating with centripetal force so taking it would make the problem complex and we just take the tip that is touching the ground on the side that is moving to. Is this correct?

if we have a cylinder then both the center of mass and the point of contact are easy to analyze so both of them work. However I have a problem if we treat the problem realistically, The object isn't rigid so it will deform a bit so it touches the ground on an area not a line. I could analyze it around the center of mass but the book says that it rotates around the last line that is touching the ground on the side that the object is moving to. I don't see how it will give the same motion, The velocity don't seem to be equal. What is wrong?

 

[anorlunda]

It sounds like you're making things more difficult than necessary.

Do you see that a small area can be approximated by a line?

Do you see that if the area is too big to be approximated by a line, and if plastic deformation is involved, you no longer have a static shape? It would take much more complicated tools, perhaps even finite element analysis, to get a really accurate answer.

We can make simplifying assumptions and use simple analytic tools to get approximate answers. Or we can use complex tools for more difficult problems. But it seems unproductive to loose sleep over the inadequacy of simple tools for complex problems.

 

[Biker]

It sounds like you're making things more difficult than necessary.

Do you see that a small area can be approximated by a line?

Do you see that if the area is too big to be approximated by a line, and if plastic deformation is involved, you no longer have a static shape? It would take much more complicated tools, perhaps even finite element analysis, to get a really accurate answer.

We can make simplifying assumptions and use simple analytic tools to get approximate answers. Or we can use complex tools for more difficult problems. But it seems unproductive to loose sleep over the inadequacy of simple tools for complex problems.

Yea, How about making the object rigid but the ground not. So there is some deformation there. The analysis will be the same as a rigid object rolling on a rigid ground, The only difference that there will be a rolling resistance torque right?

 

[Mister T]

In a rectangular shape that is tipping, Usually we take the center of mass as an axis of rotation however the center of mass is accelerating with centripetal force so taking it would make the problem complex and we just take the tip that is touching the ground on the side that is moving to. Is this correct?

I think so.

The object isn't rigid so it will deform a bit so it touches the ground on an area not a line. I could analyze it around the center of mass but [...]

The object is deformed at the contact patch, finding the center of mass is not so easy, either.

[...] the book says that it rotates around the last line that is touching the ground on the side that the object is moving to. I don't see how it will give the same motion, The velocity don't seem to be equal. What is wrong?

If the amount of deformation is small the difference is small.

Suppose you have a long metal cylinder rolling across the surface of a smooth bowl of jello. The deformation of the cylinder is negligible compared to the deformation of the jello. In the cross section shown below that line of contact is a point. Can you do the analysis?

 

[Biker]

Suppose you have a long metal cylinder rolling across the surface of a smooth bowl of jello. The deformation of the cylinder is negligible compared to the deformation of the jello. In the cross section shown below that line of contact is a point. Can you do the analysis?
View attachment 216893

I am sorry not following what exactly do you want me to do? You want me to check if last point of contact on the right gives the same velocities as if we analyze it over the center of mass? How can that point be still though?

 

[Mister T]

I am sorry not following what exactly do you want me to do? You want me to check if last point of contact on the right gives the same velocities as if we analyze it over the center of mass?

Sure. You said you wanted to figure out the dynamics part! Do you know how to use the rotational version of Newton's Second Law?

I studied statics but I thought I can figure out the dynamics part.

How can that point be still though?

It's not still. It's instantaneously at rest, like when a ball reaches the highest point after having been thrown vertically upward.
.

 

[Biker]

Sure. You said you wanted to figure out the dynamics part! Do you know how to use the rotational version of Newton's Second Law?

Yes, But just one question. If it rotates about that point, wouldn't that mean that the center of mass momentarily goes up? (if that point can withstand without deforming more)
https://i.imgur.com/nPQKVVu.png

and how do you take into account that the jello is constantly deforming while the cylinder is moving?

Sorry if I am talking non-sense :c. I thought understanding the dynamics would help.

It's not still. It's instantaneously at rest, like when a ball reaches the highest point after having been thrown vertically upward.

That is what I meant sorry.

 

[Mister T]

Yes, But just one question. If it rotates about that point, wouldn't that mean that the center of mass momentarily goes up? (if that point can withstand without deforming more)

Can you find an expression for the net torque about that point?

 

[Biker]

About rolling resistance...

In statics we took that if you have for example a rectangular shape object The resultant of the normal force will be shifted x to one side (Because all of the normal forces are pointing up but different magnitudes) and the frictional forces all point to one side so we just sum them up.However, Here I have normal forces point up tangentially to the surface of the cylinder with different magnitude plus I have frictional forces pointing perpendicular to the tangent at a point and pointing in the other direction of the rotational motion.

In one site (ref below), They place the resultant forces on a point on the perimeter. Why is that? Why it must be on the perimeter?

Sorry again asking a lot of questions :c. Is there is a good book about this point?