Mechanical advantage

Mechanical advantage quantifies how much a tool, machine, or mechanical system multiplies an applied force. This amplification comes at a cost: the input force must be applied over a greater distance than the output force moves. The underlying principle is the law of the lever. An ideal mechanism transmits power perfectly—it has no internal power source, no friction, and its parts are rigid and do not wear or deform. Real machines are compared to this ideal using efficiency factors, which account for losses.

Levers work by pivoting a rigid bar on a fixed fulcrum. Forces applied at different distances from this pivot produce different effects. The fulcrum's position determines the lever's class. When a lever rotates continuously, it acts as a rotary second-class lever, and its endpoint traces a fixed path, allowing mechanical energy to be transferred. Modern examples include gears, pulleys, and friction drives in power transmission systems. Mechanical advantage is often "collapsed" using multiple gears in a gearset, where smaller gears with lower inherent advantage are used; a "true length" rotary lever is needed for non-collapsed advantage.

As a lever rotates, points farther from the fulcrum move faster than those closer in. Since power (force times velocity) is constant throughout an ideal lever, a faster-moving point must experience a smaller force. If distances from the fulcrum to points A (input) and B (output) are *a* and *b*, the velocity ratio is *a*/*b*. The mechanical advantage (MA) is then the output force divided by the input force, equal to *a*/*b*. This is the law of the lever, which Archimedes derived geometrically. If *a* is greater than *b*, the lever amplifies force; if *a* is smaller, it reduces force. Archimedes is famously credited with saying, "Give me a place to stand and with a lever I will move the whole world." Using velocity in a static lever analysis is an application of the principle of virtual work.

For any ideal mechanism, the constant power condition gives a simple relation between mechanical advantage and speed ratio. For a gear train, input power is torque times angular velocity (P = T_A ω_A). Since output power equals input power, T_A ω_A = T_B ω_B. Thus, mechanical advantage (output torque over input torque) equals the input angular velocity divided by the output angular velocity (MA = T_B/T_A = ω_A/ω_B). This holds for all mechanical systems, from robots to linkages.

In gear trains, teeth are designed so that the number of teeth on a gear is proportional to its pitch circle radius, and meshing gears roll without slipping. The speed ratio for a pair of gears can be found from the ratio of their pitch circle radii or the ratio of their tooth counts (the gear ratio). The velocity at the contact point on the pitch circles is the same for both gears, given by v = r_A ω_A.