How it Works
A Differential in a Car

D. Indumathi, The Institute of Mathematical Sciences, Chennai

A car engine generates power which provides both (rotational) speed and torque. Torque is the turning force that is used to turn the wheels.
When you ride a cycle, you may have noticed that it is difficult to start off, but once you start moving, everything is smooth. The "starting trouble" is due to a large static friction. You need to apply more force to start the cycle moving. Once you are moving, there is only rolling friction to overcome and you do not have to apply so much pressure.

How a car engine works
The same thing happens in a car. A car engine generates power by burning petrol. A car engine may have 4 or 6 cylinders. Pistons move up and down the cylinder. When a mixtures of fuel and air burns in the combustion chamber, the resulting hot air expands and pushes the piston down. This in turn moves the crankshaft. The crankshaft converts the up and down motion of the piston into a rotatory movement that allows the wheels to turn so the car moves. 
The pistons in the cylinders are pumping up and down at high speeds—about 10-20 times a second. Even when the car is simply idling by the roadside, the pistons still need to push up and down roughly 1000 times a minute or the engine will cut out. In other words, the engine has a minimum speed at which it works best of about 1000 rpm (revolutions per minute). But that creates an immediate problem because if the engine were connected directly to the wheels, they'd have a minimum speed of 1000rpm as well—which corresponds to roughly 120km/h!
Put it another way, if you switched on the ignition in a car like this, your wheels would instantly turn at 120kmph! But there is a problem. It takes a massive amount of force to get a car moving from a standstill and an engine that tries to go at top speed, right from the beginning, won't generate enough torque to do it. To begin with, a car needs a huge amount of torque and very little speed to get it moving, so the driver uses a low gear. In effect, the gearbox is reducing the speed of the engine greatly but increasing its torque in the same proportion to get the car moving. This is because the power generated by the engine equals the product of torque and the angular speed (=2 pi rpm). Once the car starts to move, the driver switches to a higher gear. More of the engine's power is converted into speed—and the car goes faster.
So much for gears. What about the differential? 
Now you've started the car and it is smoothly moving ahead. You find that you now have to turn left (or right). What happens? When a car turns a corner, one wheel is on the "inside" of a turning arc, and the other wheel is on the "outside." Consequently, the outside wheel has to turn faster than the inside one in order to cover the greater distance in the same amount of time. You may have experienced this while practising "march-past" when turning 4-in-a-row. The person on the outside of the curve has to take really large steps and the person on the inside has to take very small steps in order to keep together. But a car is a rigid object. How to achieve this?
This is exactly where the differential comes into play. A differential is a combination of gears that allows wheels to rotate at different speeds when required, for example, when the car turns. Conversely, the differential must rotate the two wheels at the same speed when the car is moving in a straight line. This prevents the inner wheel from skidding and wearing out, trying to slow down enough to stay with the outer wheel.
For those interested in knowing more, a beautiful 10 minute You-Tube video made in 1937 is available at https://www.youtube.com/watch?v=yYAw79386WI