GEAR

Gears are wheels with toothed edges that rotate on an axle or shaft. The teeth of one gear fit into the teeth of another gear. This lets one gear turn the other, meaning one axle or shaft can be used to turn another shaft. Rotation and transmission of forces by gears. As one gear turns, the other gear must also turn. Where the gears meet, the teeth must both move in the same direction. In the diagram, the teeth of both gears move upwards. This means that the gears rotate in opposite directions. The forces acting on the teeth are identical for both gears, but their moments are different: If a larger gear is driven by a smaller gear, the large gear will rotate slowly but will have a greater moment. For example, a low gear on a bike or car. If a smaller gear is driven by a larger gear, the larger gear will rotate quickly but will have a smaller moment. For example, a high gear on a bike or car.

First, we note the geometric relationship that results from the path that the arc lengths along their circumference must be equal as the gears turn.

Since the arc lengths (shown with a heavy blue line) must be equal

r₁?₁ = r₂?₂ = arc length

If we had defined ?₂ in the opposite direction, this expression would have a negative sign (r₁?₁ = -r₂?₂).

We derive a second relationship from a torque balance. Before we can do so we must define a force between the gears termed a “contact force.” This force must be equal and opposite across the interface between the two gears, but its direction is arbitrary. We start by drawing free body diagrams with a contact force where the gears meet.

The contact force is tangent to both gears and so produces a torque that is equal to the radius times the force.

We can do a torque balance on each of the two gears

Gear 1		Gear 2

We are not usually interested in f_c, so we remove from the equations and we get

It is easy to show that the same relationship exists even if the contact forces are defined in the opposite direction (up on gear 1, down on gear 2).