How Do Rack and Pinion Gears Translate Motion in Physics?

[bhnh]

Homework Statement

I've never studied Physics, and I've run into a Physics question while writing a code for a 3D plugin. I'm making a routine which automatically creates gear trains. I've covered spur, helical and bevel gear ratios and everything works nicely, However, I've run into a brick wall with rack & pinion gears.

I know just enough to assume that this is a question of finding a ratio between the angular motion of the pinion and the linear motion of the rack. I also suspect that the angular increments have to be in radians rather than degrees. Can anyone tell me if there's a basic equation tranlating motion between these that I can incorporate into my code? Many thanks in advance.

Homework Equations

The Attempt at a Solution

 

[blochwave]

This isn't something I know offhand, but it doesn't seem too complicated(I went to the wiki page for rack and pinion to see what it looked like and am basing this off that cute .gif >_>)

Well I assume you know the rpm of the round gear, so you can find its angular velocity(w) in radians/second (and you sounded familiar with radians so I won't go into that)

So w=v/r, where r is the radius of the round gear, and v is its tangential velocity on the edge. I believe that v will be the velocity with which the flat gear is moving

If w is in radians/second and r is in meters, v will be in meters/second

Edit: So what exactly are you looking for? The ratio of angular velocity of the round gear to linear motion of the flat gear is w/v=1/r, I think. If that's what you're looking for

 

[bhnh]

Thank you, blochwave. You've certainly pointed me in the right direction. In terms of what I'm specifically after, I want the the 3D user to create a pinion gear and rack (the plugin I'm writing spaces the cogs correctly), then (a) when the pinion is rotated the rack will move the appropriate distance, and (b) when the rack is move laterally the pinion will rotate the appropriate amount.

 

[blochwave]

Oh ok, so I gave you stuff involving velocities and you want distance

When talking about "distance" with the pinion(whoo my vocab expanded)you'd say "it rotates x radians" and this would certainly most conveniently be expressed in radians because...

the length of the arc is going to be the angle in radians multipled by the radius of the circle(such is basically the definition of a radian)

So basically http://en.wikipedia.org/wiki/Image:Radian_cropped_color.svg

That's a picture of the definition of a radian, in general s=r*theta where theta is the angle in radians, r is the radius of the circle in whatever unit of distance, and s is the length of that arc in the same units of distance(you might ask how distance*radians=distance, but radians is a "dimensionless unit")

That length of arc I believe corresponds directly to the distance moved by the rack(so if the distance is d, s=d)

So you say "my pinion of radius = 2 meters(huge I know) rotates 90 degrees, how far does a point on the rack get displaced?(ie how far does it move)"

so 90 degrees=pi/2 radians, pi/2*2meters=pi meters, so the rack moved about 3.14 meters

What if the rack moves like 4 meters? so d=s=theta*r, so 4 meters=theta*2 meters

theta = 2 radians, which you can convert to degrees

 

[bhnh]

Great! Thanks, blochwave. For some reason I wasn't notified of your latest post.