Pulley & Wheel-Axle

Calculate the required effort force for wheel-axle, movable pulley, and differential pulley systems

Inputs

W

Load

D

Large Diameter

Diameter of the large (wheel) cylinder $D$

d

Small Diameter

Diameter of the small (axle) cylinder $d$; must be less than $D$

Formula Interpretation

Pulley System Diagram

Wheel-Axle

F = W \cdot \dfrac{d}{D}

The wheel-axle is a lever rotating about a common axis. The mechanical advantage equals $D/d$ . For example, with $D = 300\,\text{mm}$ and $d = 100\,\text{mm}$ the operator needs only $1/3$ of the load weight.

Movable Pulley

F = \dfrac{W}{2^n}

Each movable pulley halves the required effort; n pulleys together give a mechanical advantage of $2^n$ . The trade-off is that the operator must pull $2^n$ times as much rope as the load rises.

Differential Pulley

F = W \cdot \dfrac{D - d}{2D}

The differential pulley (chain hoist) uses two fixed sprockets of different sizes on a common axle. A very large mechanical advantage is achieved when $D - d$ is small. The textbook formula gives $F = W(D-d)/(2D)$ .

Knowledge Points

Fixed vs. Movable Pulleys

A fixed pulley is anchored to a support; it only redirects the force without changing its magnitude, so the mechanical advantage is 1. A movable pulley rises with the load; each one doubles the mechanical advantage at the cost of doubling the rope travel.

Wheel-Axle Principle

The wheel-axle is equivalent to a continuous lever: the effort is applied at radius D/2, the load hangs at radius d/2, and by the law of the lever F·(D/2) = W·(d/2). The mechanical advantage D/d can be made arbitrarily large by increasing the ratio of wheel to axle diameter.

Differential Pulley (Chain Hoist)

The differential block works by having the effort rope wind onto the large sprocket (D) while the load rope unwinds from the small sprocket (d). Each full revolution of the top block raises the load by (D−d)/2 while consuming πD of rope travel. This gives a mechanical advantage of 2D/(D−d), which becomes very large when D ≈ d.

Worked Examples

Example ① — Wheel-Axle

A wheel-axle has $D = 300\,\text{mm}$ and $d = 100\,\text{mm}$ . Find the force needed to lift $W = 1200\,\text{N}$ .

F = 1200 \times \dfrac{100}{300} = 400\,\text{N}

\text{MA} = \dfrac{300}{100} = 3

Example ② — 3 Movable Pulleys

A system of $n = 3$ movable pulleys is used to lift $W = 800\,\text{N}$ .

F = \dfrac{800}{2^3} = \dfrac{800}{8} = 100\,\text{N},\;\text{MA} = 8

Example ③ — Differential Pulley (from textbook)

A differential pulley has $D = 300\,\text{mm}$ and $d = 250\,\text{mm}$ . Find the force to lift $W = 5000\,\text{N}$ .

F = 5000 \times \dfrac{300 - 250}{2 \times 300} = 5000 \times \dfrac{50}{600} = 417\,\text{N}

\text{MA} = \dfrac{2 \times 300}{300 - 250} = \dfrac{600}{50} = 12

Summary: differential pulley gives the greatest mechanical advantage (≈12×) with a compact mechanism; 3 movable pulleys give 8×; the wheel-axle gives 3× in this example.

Extended Knowledge

•In practice, real pulleys have friction losses. The actual effort required is F_actual = F_ideal / η, where η is the efficiency (typically 0.7–0.95 for wire-rope systems). Greased chain hoists can achieve η ≈ 0.90.
•A block-and-tackle can combine multiple movable and fixed pulleys. The general rule is that if k rope segments support the load, then F ≈ W/k (ignoring friction). Counting the supporting rope segments is the fastest way to find the mechanical advantage.
•The differential pulley principle is used in Weston differential chain blocks, commonly used in automotive workshops and construction to lift engines and structural members. The self-locking property (the load does not descend when the effort is released) makes them inherently safe for overhead lifting.