How Do Wheels Work? The Mechanics of Axles and Wheels

Wheels are everywhere in our modern technological society, but they have also been used since ancient times. The place you are most likely to see a wheel is on a vehicle or trailer, but wheels are used for a variety of other applications. They are widely used in machines in the form of gears, pulleys, bearings, rollers and hinges.

Before you read this article which becomes a bit technical, it would be helpful to read another related hub which explains the basics of mechanics.

Wheels were unlikely to have been invented by just one person, and probably developed in many civilizations independently over the millenia. We can only imagine how it happened. Maybe some bright spark noticed how easy it was to slide something over ground with rounded stone pebbles on it, or observed how easily tree trunks could be rolled, once cut down. The first "wheels" were probably rollers made from tree trunks and positioned under heavy loads. The problem with rollers is that they are long and heavy and have to be continually re-positioned under the load, so the axle had to be invented to hold a thinner disk, effectively a wheel, in place. Early wheels were likely made from stone or flat boards joined together into the form of a disk.

It all boils down to reducing friction. So imagine if you have a heavy weight resting on the ground. Newton's 3rd Law states that "For every action, there is an equal and opposite reaction". So when you try to push the load, that is the action. The corresponding reaction is the force of friction acting backwards and is dependent on both the nature of the surfaces in contact and the weight of the load. This is known as static friction or stiction and applies to dry surfaces in contact. Initially the reaction matches the action in magnitude and the load doesn't move, but eventually if you push hard enough, the friction force reaches a limit and doesn't increase further. If you push harder, you exceed the limiting friction force and the load starts to slide. The force of friction however continues to oppose motion (it reduces a bit once motion starts), and if the load is very heavy and/or the surfaces in contact have a high coefficient of friction, it can be difficult to slide it.
Wheels eliminate this friction force by using leverage and an axle. They still need friction so that they can "push back" on the ground on which they roll, otherwise slippage occurs. This force however doesn't oppose motion or make it more difficult for the wheel to roll.

The lever is one of the simplest machines. Levers are great because they give a mechanical advantage. This means that they can produce a greater force than that which you apply to them. You have used a lever in some shape or form without actually realizing it. So for instance scissors, nut crackers, pliers, hedge shears, bolt cutters and lopping shears all employ levers in their design. A prybar or crowbar is a lever also, and when you prise open the lid of a tin with the handle of a spoon, you are using "the law of the lever" to create a greater force. A long handle on a wrench provides more "leverage". A claw hammer also acts as a lever when pulling out nails. A see-saw is also a lever.

To understand how wheels work, we first need a basic understanding of how levers work and the science behind them. In the diagram below, two forces act on the lever. This is a schematic or diagram, but it symbolically represents any of the real life levers mentioned above.

The lever pivots at a point called a fulcrum represented by the black triangle (in real life, this could be the screw holding the two blades of a scissors together). A lever is said to be balanced when the lever doesn't rotate and everything is in equilibrium (e.g. two people of equal weight sitting on a see-saw, at equal distances from the pivot point).

In the diagram above, two forces F1 and F2, at distances d1 and d2 respectively, act on the lever.

When balanced:

"The sum of the clockwise moments equals the sum of the counter-clockwise moments"

The moment of a force about a point is determined by multiplying the magnitude of the force by the distance from the point.

So for F1, the moment is F1d1

and for F2, the moment is F2d2

And when the lever is balanced, i.e. not rotating and static:

F1d1 = F2d2

Imagine if F1 is the active force and is known. F2 is unknown but must push down on the lever to balance it.

Rearranging the equation above

F2 = F1(d1/d2)

So F2 must have this value to balance the force F1 acting down on the right hand side.

Since the lever is balanced, an equivalent force, equal to F2, shown in orange must be pushing upwards on the left side of the lever.

If the distance d2 is a lot smaller than d1 (which would be the case with a crowbar or pliers), the term (d1/d2) in the equation above is greater than unity and F2 becomes greater than F1. (a long handled crowbar can easily produce a ton of force).

This is intuitively correct since we know how a long crowbar can create a lot of force for lifting or prying things, or if you put your fingers between the jaws of a pliers and squeeze, you know all about it!

If F2 is removed and the lever becomes unbalanced, the upwards force due to the force F1 on the right is still F1(d1/d2). This force magnifying effect or mechanical advantage of a lever is one of the features that makes it so useful.

This analysis applies to the example above where the wheel is subject to a force or effort F at the axle.

Fig. 1

A force acts on the axle whose radius is d.

Fig. 2

Two new equal but opposite forces are introduced where the wheel meets the surface. This technique of adding fictitious forces which cancel each other out is useful for solving problems.

Fig. 3

When two forces act in opposite directions, the result is known as a couple and its magnitude is called the torque. In the diagram, the added forces result in a couple plus an active force where the wheel meets the surface. The magnitude of this couple is the force multiplied by the radius of the wheel.

So Torque T_w = Fd.

Fig. 4

A lot is going on here! The blue arrows indicate the active forces, the purple the reactions. The torque T_w which replaced the two blue arrows, acts clockwise. Again Newton's third law comes into play and there is a limiting reactive torque T_r at the axle. This is due to friction caused by weight on the axle. Rust can increase the limiting value, lubrication reduces it.

Another example of this is when you try to undo a nut which is rusted onto a bolt. You apply a torque with a wrench, but the rust binds the nut and acts against you. If you apply enough torque, you overcome the reactive torque which has a limiting value. If the nut is thoroughly seized up and you apply too much force, the bolt will wring.

In reality things are more complicated and there is additional reaction due to the moment of the inertia of the wheels, but let's not complicate things and assume the wheels are weightless!

• The weight acting down on the wheel due to the weight of the cart is W.
• The reaction at the ground surface is R_n = W
• There is also a reaction at the wheel/surface interface due to the force F acting forwards. This doesn't oppose movement but if it is insufficient, the wheel won't turn and will slide. This equals F and has a limiting value of F_f = uR_n.

Fig. 5

The two forces which produce the torque T_w are shown again. Now you can see this resembles a lever system as explained above. F acts over distance d, and the reaction at the axle is F_r.
The force F is magnified at the axle and is shown by the green arrow. Its magnitude is:

F_e = F(d/a)

Since the ratio of the wheel diameter to the axle diameter is large , ie d/a, the minimum force F required for movement is proportionately reduced. The wheel effectively works as a lever, magnifying the force at the axle, and overcoming the limiting value of the friction force F_r. Notice also for a given axle diameter a, if the wheel diameter is made bigger, F_e becomes larger. So it's easier to push something with big wheels than small wheels because there is a greater force at the axle to overcome friction.

Since

Torque = Force at Axle x Radius of Wheel

for a given force at the axle, the torque acting at the axle is bigger for bigger wheels. So friction at the axle is greatly overcome,and therefore it's easier to push something with bigger wheels. Also if the surface that the wheel rolls on isn't very flat, larger diameter wheels tend to bridge imperfections, which also reduces the effort required.

When a wheel is driven by an axle, since

Torque = Force at Axle x Radius of Wheel

therefore

Force at Axle = Torque / Radius of Wheel

So for a constant driving torque, smaller diameter wheels produce a greater tractive effort at the axle than larger wheels. This is the force which pushes a vehicle.