How does a wheel and axle provide mechanical advantage?

The wheel and axle is a simple machine designed to amplify force. It operates on the principle that a force applied to a larger radius can overcome a greater force acting at a smaller radius.12

When used for tasks like lifting weights, the system typically employs drums of different diameters with ropes wrapped around them. The mechanical advantage is determined by the ratio of the radius of the large drum ($R$) to the radius of the small drum ($r$). Mathematically, this is expressed as:2

$$\text{Mechanical Advantage}=\frac{R}{r}$$

In this arrangement, an operator pulling the rope on the large drum exerts a force ($F$) that is sufficient to lift a larger weight ($W$) attached to the rope on the small drum. While the theoretical mechanical advantage is defined by the ratio of these radii, the actual mechanical advantage is often lower due to the effects of friction. By adjusting the radii of the components, such as using specialized pulley configurations, the system can achieve significant force amplification.2