Rolling Gear and Rack Question

[Zalajbeg]

Hi everyone,

http://www.freeimagehosting.net/newuploads/wt15k.jpg

In the picture there is a rack at the bottom and there is a gear above it. The gear is on a axis which can move on a horizontal line and the gear can rotate without friction between the axis and the horizontal housing.

1) If the rack is pulled to the right by a distance of x, will the gear only go to the right without rotation or will it also rotate?

2) If it rotates is there a certain ratio between the distance x and the number of revolutions or does it depend on the velocity of the rack?

 

[Naty1]

assuming the gear and 'line' are not weightless, and there is some friction between the gear teeth, it will turn as does a typical gear. Other assumptions would yield different results.

The distance traveled and the number of revolutions are related via the number of teeth...one revolution = # teeth which = C = 2[pi]r...

one revolution equals C in horizontal distance.

 

[Zalajbeg]

assuming the gear and 'line' are not weightless, and there is some friction between the gear teeth, it will turn as does a typical gear. Other assumptions would yield different results.

The distance traveled and the number of revolutions are related via the number of teeth...one revolution = # teeth which = C = 2[pi]r...

one revolution equals C in horizontal distance.

I wonder if the rack will not also carry the gear horizontally with its inertia apart from the displacement because of the rotational motion.

 

[AlephZero]

The gear teeth make this more complicated and confusing. It's simpler to get started by assuming there is a wheel that rolls on the "rack" without slipping.

If you draw a free body diagram for the forces on the wheel, then when the rack accelerates it should be obvious the wheel also accelerates.

The center of the wheel will also move, but the amount it moves depends on the mass and the radius of gyration of the wheel. If the mass of the wheel is all at the center or all around the rim, the motion willl be different.

If the gear teeth are cut to a well designed shape (not the triangles you drew) there will be no "friction" between the gears, because the surfaces of the teeth don't "slip" over each other but "roll" over each other just like the wheel rollinig without slipping. If the gear teeth do "slip", and there is friction between them, analyzing what happens in detail would get VERY complicated.

 

[Naty1]

... but the amount it moves depends on the mass and the radius of gyration of the wheel. If the mass of the wheel is all at the center or all around the rim, the motion willl be different.

I fail to see how the mass of the wheel is anyway involved given the initial conditions posed. The rack moves distance 'x'...period.

 

[Zalajbeg]

The gear teeth make this more complicated and confusing. It's simpler to get started by assuming there is a wheel that rolls on the "rack" without slipping.

If you draw a free body diagram for the forces on the wheel, then when the rack accelerates it should be obvious the wheel also accelerates.

The center of the wheel will also move, but the amount it moves depends on the mass and the radius of gyration of the wheel. If the mass of the wheel is all at the center or all around the rim, the motion willl be different.

If the gear teeth are cut to a well designed shape (not the triangles you drew) there will be no "friction" between the gears, because the surfaces of the teeth don't "slip" over each other but "roll" over each other just like the wheel rollinig without slipping. If the gear teeth do "slip", and there is friction between them, analyzing what happens in detail would get VERY complicated.

Then I have two more questions. Let's assume the teeth are well shaped so that they roll over each other. And the rack is pulled with a constant velocity.

1) If a horizontal force is exerted to the top of the gear in the opposite direction of the motion (The force is not strong enough to stop the gear or change the direction of the motion) will the number of revolution be lower? Will the gear have to slip?

2) If the force is in the same direction with the motion will it roll faster than there is no force?

 

[engnr_arsalan]

1)suppose there is no friction between the shaft(that is holding gear)and the horizontal line.
rack is pulled to right with a force say "F" which is less then the rotational inertial effects/force of gear and shaft both.
under these assumption the gear will not rotate it will simply slide to the right.