Why Does a Wheel Travel 2*pi*r in One Revolution?

[mech-eng]

I can not establish the mathematical reason why when a wheel of radius r takes a full revolution it moves 2*pi*r. Can someone help me.

Have a nice day.

 

[olivermsun]

Well, the circumference (length measured around the outside of the wheel) is 2*pi*r, so if the wheel doesn't slip on the ground then it has to move forward 2*pi*r to complete a full revolution.

As to why the circumference of a circle C = 2*pi*r … I think this is just one of those cases where "the world just works that way." I mean, pi is just defined as the constant such that pi = C / (2r).

 

[mech-eng]

Well, the circumference (length measured around the outside of the wheel) is 2*pi*r, so if the wheel doesn't slip on the ground then it has to move forward 2*pi*r to complete a full revolution.

As to why the circumference of a circle C = 2*pi*r … I think this is just one of those cases where "the world just works that way." I mean, pi is just defined as the constant such that pi = C / (2r).

I know this but without observation how can we understand it should go as far as it is circumference after one revolution. Is there any mathematical idea for this? In other words, why should the way it goes equal to it is circumference and I want to learn that whether this is a stupid question or not? Sometimes we can not see or understand our mistakes.

 

[Drakkith]

Consider a point on the wheel. Turning the wheel one revolution causes the point to trace out a path equal to the circumference of the wheel. There's really not much else to it. I think you may be reading too far into it, lol.

 

[olivermsun]

Well, you are specifying the wheel doesn't slip, so if the wheel is touching the ground at a certain point, then the only thing that can happen when you turn the wheel is the first point stays put and then the next one touches down right next to it, and so on.

Mathematically you could map each point on the wheel to a unique point on the ground, and it would turn out to span 2*pi*r. Or you could imagine wrapping a string around the wheel and then unwinding it on the ground, and it would span 2*pi*r. There are lots of ways to imagine the problem.

 

[A.T.]

Now try this question:

Imagine two identical coins lying on a table so they are in contact. One is held fixed, while the other one rolls around the circumference of the fixed one, without slipping between the coins. How many revolutions around its own center will the rolling coin undergo before it returns to its original position?

 

[jackwhirl]

Please examine the animation here: http://en.wikipedia.org/wiki/File:2pi-unrolled.gif
That may take some of the mystery out of it.

 

[ehild]

You can prove it experimentally with a paper roll as shown in the figure. Draw a mark on the cross section of the cylinder at the end of the sheet. Put the cylinder on the table so the mark points vertically down, and keep the end of the sheet fixed while rolling the cylinder on the table, until the mark is vertically down again. A certain length of paper is unwound and that length of the piece is equal to the displacement of the centre of the cylinder. Wrap the piece of unwound sheet to the cylinder again: the end will coincide with the mark, so its length is equal to the circumference of the cylinder.

ehild