# What is the radius of a gear?

1. Feb 6, 2017

### Parsifal1

I want to find how far a gear from a rack and pinion will rotate from the displacement of the rack and the angle that it will have turned using: s=r*angular displacement.

So what is the radius in this case?

2. Feb 6, 2017

### Staff: Mentor

Can you post a diagram to make this a bit clearer please? Also, is this for schoolwork?

3. Feb 6, 2017

### Parsifal1

Its a project. I don't know which type of gear I will be using yet, I just want to know what you would use for the radius in that formula in the case of a gear. Would you use the radius of gyration, using the outer and inner radius formula?

4. Feb 6, 2017

### Nidum

Gear motion calculations are based on pitch circle diameter - diameter D in your diagram

5. Feb 6, 2017

### Parsifal1

So I couldn't use the formula s=r*angular displacement to work out the rotation of a rack and pinion gear?

6. Feb 6, 2017

7. Feb 6, 2017

8. Feb 6, 2017

### Nidum

The effective motion geometry of a rack and pinion is that of a plain cylinder of diameter equal to the gear pitch circle diameter rolling without slip against a flat surface .

9. Feb 6, 2017

### Parsifal1

So the linear displacement of the rack is equal to the linear displacement of the gear? If so I should be able to use the formula I mentioned?

10. Feb 6, 2017

### Nidum

Linear displacement of rack = ( angle turned by pinion / 360 ) x Pi x pitch circle diameter .

or

Linear displacement of rack = ( angle turned by pinion / 360 ) x Pi x 2 x pitch circle radius .

11. Feb 6, 2017

### Nidum

You can also derive formulas based directly on gear module number rather than on pitch circle diameter/radius

12. Feb 6, 2017

### Mech_Engineer

The rack's displacement is equal to the tangential displacement (arc length) of the pinion at the pitch diameter. So if the pitch diameter of the pinion is 20 mm (radius of 10 mm), and you rotate the pinion 90 degrees (π/2 radians), the rack will move about 15.71 mm (calculation here).

13. Feb 6, 2017

### Parsifal1

I see, so that was the formula I mentioned then. Problem solved.

14. Feb 7, 2017

### Baluncore

You do not need to measure the pitch circle radius. There is an easier way.
You only need to know the pitch of the linear gear rack and the number of teeth on the gear wheel.
That will give you the relationship between linear position and rotation angle.