The gearing of an automobile differential is illustrated as following in final form.

Automobile Differential Complete Schematic

Without the "square" set of four gears in the middle of the above diagram which yields to the figure below, both wheels turn at the same angular velocity. This leads to problems when the car negotiates a turn.

Automobile Wheel Drive without Differential

Now imagine the differential "square" alone, as illustrated in the following figure. It should be apparent that turning one wheel results in the opposite wheel turning in the opposite direction at the same rate.

Automobile Differential Alone

This is how the automobile differential works. It only comes into play when one wheel needs to rotate differentially with respect to its counterpart. When the car is moving in a straight line, the differential gears do not rotate with respect to their axes. When the car negotiates a turn, however, the differential allows the two wheels to rotate differentially with respect to each other.

One problem with an automotive differential is that if one wheel is held stationary, the counterpart wheel turns at twice its normal speed as can be seen by examining the complete scheme of automobile differential. This can be problematic when one wheel does not have enough traction, such as when it is in snow or mud. The wheel without traction will spin without providing traction and the opposite wheel will stay still so that the car does not move. This is the reason for a device known as a "limited slip differential" or "traction control".

The gear train in an automobile differential is a common application of gears, but often misunderstood by the lay public. Here we present a simplified explanation of how and why an automobile differential works.

The car is turning about a circle with nominal radius r_n. (For this discussion, we assume that the axis of the wheel axle for the driven (rear) wheels passes through the turn circle center. This is typically true only for a fairly large radius of turn.)

The outer wheel traverses an arc with radius r_o and the inner wheel traverses an arc with radius r_i. As illustrated, the lengths of the arcs traversed are s_o, s_n, and s_i. The outer arc s_o is obviously larger than the inner arc s_i for a given traversed angle theta. Some way of ensuring that the outer wheel is able to turn slightly faster than the inner wheel must be ensured in order to prevent binding and slippage of the tires on the road. For non-driven wheels which simply rotate freely independently of other machinery, this is not a problem. Driven wheels connected to the engine via the driveshaft, however, must both be turned by gearing and this gear train must allow for differential movement of the left wheel with respect to the right wheel. This is a difficult problem since for every turning circle the differential rotation of the left and right wheels is different. Fortunately, the automobile differential solves this problem with only one transmission and one drive shaft for both driven wheels.

Since s=r(THETA), the length of the arc traversed for a given theta is proportional to the radius. Since r_o is greater than r_n by the same amount that r_i is less than r_n, the right wheel center must travel further than the car center by the same amount that the left wheel center must travel less further than the car center. As its name implies, a differential allows the left and right drive wheels to turn differentially with respect to each other. As can be seen by turning the drive wheels of a car on a mechanic's lift, turning one drive wheel results in the opposite wheel turning at the same rate in the opposite direction.