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

  • Context: High School 
  • Thread starter Thread starter mech-eng
  • Start date Start date
  • Tags Tags
    Wheel
Click For Summary

Discussion Overview

The discussion revolves around understanding why a wheel travels a distance of 2*pi*r during one complete revolution. Participants explore the mathematical and conceptual reasoning behind this phenomenon, including the implications of wheel slipping and various illustrative examples.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the mathematical reasoning behind the distance traveled by a wheel in one revolution.
  • Another participant states that the circumference of the wheel is 2*pi*r, suggesting that if the wheel does not slip, it must move forward that distance to complete a revolution.
  • A similar point is reiterated by another participant, who questions the understanding of why the distance traveled equals the circumference, expressing a desire for deeper mathematical insight.
  • One participant describes the motion of a point on the wheel, indicating that it traces out a path equal to the circumference during one revolution.
  • Another participant elaborates on the non-slipping condition, explaining how each point on the wheel corresponds to a unique point on the ground, leading to the distance of 2*pi*r.
  • A new question is posed about the number of revolutions a rolling coin makes around a fixed coin, introducing a related but distinct scenario.
  • A participant suggests examining an animation to visualize the concept better.
  • Another participant proposes an experimental method using a paper roll to demonstrate the relationship between the unwound length of paper and the wheel's displacement, reinforcing the idea of circumference.

Areas of Agreement / Disagreement

Participants express various viewpoints and methods of understanding the distance traveled by a wheel, with no consensus reached on a singular explanation or mathematical framework. The discussion includes both agreement on the basic principles and differing interpretations of the underlying reasoning.

Contextual Notes

Some participants express uncertainty about the foundational reasoning and seek further clarification, indicating that assumptions about non-slipping conditions and the nature of circular motion may affect their understanding.

mech-eng
Messages
826
Reaction score
13
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.
 
Physics news on Phys.org
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).
 
olivermsun said:
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.
 
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.
 
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.
 
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?
 
You can prove it experimentally :smile: 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
 

Attachments

  • sheetroll.JPG
    sheetroll.JPG
    18 KB · Views: 514

Similar threads

High School Rotational Dynamics: Confused about the tension in this string pulling a mass up an incline
  • · Replies 2 ·
Replies
2
Views
2K
High School Derivation of Wheel Relationship Formulas
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad How Does Changing Mass and Spring Force Affect Watch Oscillators?
  • · Replies 11 ·
Replies
11
Views
2K
Graduate How fast will two spinning weighted wheels propel a 7 lb cylinder?
  • · Replies 19 ·
Replies
19
Views
4K
High School The physics of flywheel launchers (like tennis ball shooters)
  • · Replies 60 ·
3
Replies
60
Views
7K
High School Questions about momentum and force as well as normal force/friction
  • · Replies 35 ·
2
Replies
35
Views
5K
Undergrad Inclined plane and 2 wheels problem
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Car Accident Physics - Are Wheels/Tires Good or Bad Energy Absorbers?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Finding revolution of different sized wheels
  • · Replies 4 ·
Replies
4
Views
5K
Undergrad Finding Moment of Inertia of a wheel experimentally
  • · Replies 4 ·
Replies
4
Views
2K