Relative Motion. Wheel and axle.

[fisico]

This is the PROBLEM:

Flinstone has a problem too. He built a beam mover, described below (on the attachment, DOC file, look down), but can't figure it out. Perhaps you can help. Two identical wheels and axles sets were carved from from logs. In each set, the 50 cm diameter wheels and the 14 cm diameter axle were a single piece of wood. They are used to move a very long wooden beam resting on the axles as shown. If the beam moves 80 cm over the level ground, how far does the center of each wheel move? ANS = 64 cm

I know that if the wheel moves one revolution 2pi*D, then the center moves pi*D, but the beam also moves over the axle. I'm not sure if here the beam moves
2pi*d or pi*d.

where D is the radius of the wheel and d is the radius of the axle.

Any help would be great. Thanks a lot.

 

[tony873004]

1 revolution is not 2pi*D if D is diameter.

 

[andrewchang]

it says that D is the radius

 

[Doc Al]

I know that if the wheel moves one revolution 2pi*D, then the center moves pi*D,

You might want to rethink this statement.

 

[fisico]

I know I shoud have used R instead of D to talk about the radius. It was stupid. Anyways,

Doc Al, if you say that's wrong then tell me why and justify your comment. If you can help or want to then please respond accordingly. Thanks.

 

[Doc Al]

You seem to think that if a wheel (assumed to be rolling without slipping along the ground) rolls through one revolution (a distance X) that its center only moves half as much (a distance X/2). Why do you say that?

Try it and see! Take a soda can (or whatever) and roll it along a piece of paper. When the can rolls through one cirumference, how fall does the can move?

The way I would look at this problem is as follows. Assume the wheel is rolling at some rate. How fast does the center move with respect to the ground? How fast does the top of the axle move with respect to the ground? (Realize that the beam moves at the same speed as the top of the axle.) Compare these speeds to compare the distance each moves.

 

[fisico]

http://www.phys.unsw.edu.au/~jw/rolling.html

I will for sure try that, but first you should look at the address above.( good explanation, but my question above remains) The wheel does move faster than the axle for sure. When driving in a car just get your head out and look at the wheel and then at the center of the wheel. With respect to your view the center of the wheel is not moving but the wheel definitely does. This type of subject should be easy but I'm not receiving enough feedback, maybe it isn't?

 

[Doc Al]

http://www.phys.unsw.edu.au/~jw/rolling.html

I will for sure try that, but first you should look at the address above.( good explanation, but my question above remains)

That website confirms what I just explained above. Just study the first diagram: The wheel has a diameter of 2 units and thus a circumference ($2 \pi$) of about 6.3 units. Note that the center of the wheel moves about that distance as it rolls through one complete turn.

The wheel does move faster than the axle for sure. When driving in a car just get your head out and look at the wheel and then at the center of the wheel. With respect to your view the center of the wheel is not moving but the wheel definitely does.

Realize that the wheel rotates around the center. Looked at from the viewpoint of the car, the top of the wheel moves forward while the bottom of the wheel moves backward. And, with respect to the car, the center doesn't move at all.

But look at it from the viewpoint of someone on the ground. The instantaneous speed of each part of the wheel is different. The bottom of the wheel doesn't move at all with respect to the ground, while the top moves twice as fast as the center. (I think that's what you were thinking when you said that the center of the wheel moves "half the distance". But you were confusing the instantaneous speed of the top of the wheel with the speed at which the wheel covers ground.)

 

[fisico]

THANKS SO MUCH! It all looks so much clear for me now. Both horizontal and vertical components are concentrated as one at the top of the wheel while otherwise it is distributed in two dimensions, always conserving the total amount. It has to be so since per assumption kinetic energy is conserved. But the horizontal component is the one we are looking at here are we?

So, let me see if I get it right, for the actual problem, taking into account what you say about the distance covered by the wheel and center which are the same, (and I was totally blind not to see on that website) and not talking about the constants pi or 2pi.

Let the radius of the wheel be R and radius of axle be r. If the center of the wheel moves R then the beam moves R + r. In other words, the beam moves what the center does plus what the axle moves.

If it is so, then the center moves a fraction of what the beam does:

R/(R+r) * 80 cm = 64 cm, which is the actual answer.

Is anything above right? Anyways I really appreciate your comments. Thanks a lot.

 

[Doc Al]

Exactly right!

 

[boit]

Doesn't the above discourse prove that a wheel bearing a load on the axle is actually a second class lever. Picture this: I have to transport a rim to point b. Rather than separate it from the tire, i decide to tap the top of the tire as i move along. If the tire is twice the diameter of the rim, am i not getting a mechanical advantage of approx. 2? Is this not a second class lever? The contact point at the surface act as the instantaneous fulcrum. The opposite end, where am tapping, is the effort arm god the load is in between. Not different from a wheelburrow, only better as far as friction and lubrication is concerned.