Radius of a Gear: Rotating Disk Homework

[the_dialogue]

Homework Statement

Suppose a disk of radius 'r' is rotation on a surface. If the center G moves a distance 'd', then what is the distance traveled by a point on the top of the disk (on its edge or circumference).

Homework Equations

theta = s / r ; where theta is the rotation in rad, s is the arc length, r is the radius

The Attempt at a Solution

I know that if G moves a distance 'd', then the entire circle rotates 'theta'=d/r. But I'm not sure how to make this a general case.

A thought: Can i treat the point of contact between the disk and ground as a n "origin" and then state that a point directly above it on the edge of the disk moves '2r*theta' ?

Thank you,
Alex.

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EDIT:

I hope you don't mind if I make the problem a bit more specific. Suppose a gear of radius r_o is moving with an inner hub of radius r_i. If I know the origin moves 'd', then how far does a point on the circumference of the inner hub move?

 

[OlderDan]

It is not clear to me what you are asking. First, are you talking about a disk that is rolling in a straight line on it edge? I think so, but need to be sure. Are you looking for the net displacement of the point on the edge, or its actual path length? Are you looking at all points, or only the point that started at the top. By changing the problem with your edit, you seem to be generalizing to any point on the wheel. The answer is not the same for all points, whether you are talking about displacements or path lengths. Please restate the problem being specific about the orientation of the disk and what exactly you are trying to calculate.

 

[BobG]

Homework Statement

Suppose a disk of radius 'r' is rotation on a surface. If the center G moves a distance 'd', then what is the distance traveled by a point on the top of the disk (on its edge or circumference).

Homework Equations

theta = s / r ; where theta is the rotation in rad, s is the arc length, r is the radius

The Attempt at a Solution

I know that if G moves a distance 'd', then the entire circle rotates 'theta'=d/r. But I'm not sure how to make this a general case.

A thought: Can i treat the point of contact between the disk and ground as a n "origin" and then state that a point directly above it on the edge of the disk moves '2r*theta' ?

Thank you,
Alex.

Are you talking about a wheel rolling along the ground?

You want the center of the wheel to be your origin. You find the horizontal component (x-component) of the point on the circumference relative to the origin. It would be best to pick the trailing edge of the wheel as your point of interest. That way, it's start point would be -r.

As the wheel rotates, find the point's x component relative to the center. You should be able to find a general equation that would handle any location around the circumference, plus be correct for your start position. Your equation $$\theta = \frac{s}{r}$$ is on the right track, but you need to find out just the horizontal displacement.

Add in the distance that the center of the wheel moved.

Subtract your start position from the above sum.