Wheel and Axle

Coaxial wheel and axle; mechanical advantage, torque, efficiency and the golden rule.

The wheel and axle is a simple machine that converts a small effort applied at a large radius into a large resistance developed at a small radius, or vice versa. Steering wheels, screwdrivers, water-tap handles, winches, and flywheels are all examples of the same principle: two coaxial cylinders of different radii that rotate together, exchanging force for arc travel.

Conceptual context

The physics of the wheel and axle is the physics of torque. When a force acts at a certain distance from a rotation axis, it produces a rotational effect proportional to the product of force times distance. Two forces in equilibrium on the same machine must generate equal and opposite torques about the common axis. That torque-balance principle is the seed of all the formulas in this leaf.

From an energy perspective, the wheel and axle is a pure exchanger: what you gain in force you pay in arc length traveled. The input work $F_R$ times $s_R$ is always greater than or equal to the useful work $F_r$ times $s_r$, the difference being energy dissipated by friction. This is the Golden Rule applied to rotation.

🟢 Essential level

The core idea is simple: if the wheel is large and the axle is small, a force applied at the wheel can overcome a much larger force at the axle, because the wheel arm $R$ is greater than the axle arm $r$.

Imagine a stiff water tap. If you push directly on the metal spindle, you need a lot of force. If you fit a long handle, the same torque is achieved with much less force. The handle is the wheel; the tap spindle is the axle.

In the ideal model, the effort $F_R$ you apply at the wheel is related to the resistance $F_r$ at the axle through the ratio r/R. Since R is greater than r, we have $F_R$ less than $F_r$: you apply less force than you get.

The ideal mechanical advantage $VM$ summarizes that gain.

If $R$ is 30 cm and $r$ is 3 cm, the mechanical advantage is 10: with 50 N at the wheel you can overcome 500 N at the axle. The price is that the hand travels 10 times more arc than the load point.

🔵 Formal level

Basic formulas

The key result in the ideal model is:

Ideal system effort

The ideal mechanical advantage $VM$ is:

Ideal mechanical advantage

The base expression for rotational moment is:

Torque or moment of force

These ideal relations use the two radii and the applied force to predict how much effort the mechanism can transmit before frictional losses are introduced.

Real student questions

• Why can't I make the axle infinitely small for unlimited advantage? The axle must withstand the output torque $F_r$·$r$. If $r$ decreases, shear stress in the axle material increases, eventually leading to mechanical failure.
• Does the wheel and axle work as a speed multiplier? Yes — if effort is applied at the axle and resistance at the wheel, force is lost but speed is gained. A bicycle pedal-to-rear-sprocket system works this way.
• What is the difference between a wheel-and-axle and a gear? In both, torque is transmitted between elements of different radius. In the wheel and axle the two cylinders rotate together on the same axis; in gears, torque passes between two separate axes via meshing teeth.

Final synthesis

The wheel and axle transforms rotation at a large radius into torque at a small radius. The mechanical advantage $VM$ is R/r — the machine's identity card; $\eta$ is the machine's performance card. Together they determine the real force the user obtains per Newton applied at the wheel.