# Radius of a Gear: Rotating Disk Homework

• the_dialogue
In summary, the distance traveled by a point on the top of a disk of radius 'r' that is rotating on a surface, when the center G moves a distance 'd', can be calculated by finding the horizontal component of the point on the circumference relative to the origin, adding the distance the center moved, and subtracting the start position of the point. This equation can be applied to any point on the circumference and will result in the correct displacement.
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?

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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.

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the_dialogue said:

## 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.

## 1. What is the radius of a gear?

The radius of a gear is the distance from the center of the gear to the outer edge of the gear. It is typically measured in millimeters or inches.

## 2. How do you calculate the radius of a gear?

The radius of a gear can be calculated by dividing the diameter of the gear by 2. The diameter is the distance across the widest part of the gear.

## 3. Why is the radius of a gear important?

The radius of a gear is important because it affects the speed and torque of the gear. A larger radius will result in a higher speed and lower torque, while a smaller radius will result in a lower speed and higher torque.

## 4. Can the radius of a gear be changed?

Yes, the radius of a gear can be changed by adjusting the size of the gear or by using different gears with varying radii. This can be done to achieve different speeds and torques in a gear system.

## 5. How does the radius of a gear affect the gear ratio?

The gear ratio is directly affected by the radius of a gear. A larger gear radius will result in a higher gear ratio, while a smaller gear radius will result in a lower gear ratio. This relationship is important in determining the overall function and efficiency of a gear system.

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