The law of the lever: torque balance and mechanical advantage
Where the balance equation comes from, how the three lever classes differ, and how effort and load arms trade force for distance.
Torque is what actually balances
A lever pivots about a fulcrum, and what balances around that pivot is torque, the product of a force and its distance from the pivot. When the clockwise torque from the load equals the counter-clockwise torque from your effort, the lever holds steady. Writing that equality out gives effort force times effort arm equals load force times load arm, the single relationship this calculator solves. Everything else, including mechanical advantage, falls out of that balance.
Trading force for distance
A lever never gives you something for nothing; it trades force against distance. A long effort arm and a short load arm let a small push move a large load, but your end of the lever has to travel farther than the load does. The ratio of those travels equals the mechanical advantage, so a lever with an advantage of four moves the load a quarter as far as your hand moves. This is why a crowbar with a long handle lifts a heavy crate through only a small rise.
The three classes of lever
First class levers put the fulcrum between the effort and the load, like a seesaw or a pair of scissors, and can multiply force or speed depending on arm lengths. Second class levers place the load between the fulcrum and the effort, like a wheelbarrow or a nutcracker, and always give a mechanical advantage above one. Third class levers place the effort between the fulcrum and the load, like tweezers, a fishing rod, or your forearm, and give an advantage below one in exchange for greater range and speed at the load end. The balance law is identical for all three; only the layout changes.
Worked example you can reproduce
Say a load of 200 N sits 0.5 m from the fulcrum and you can only reach 2 m out on the other side. Setting effort times 2 equal to 200 times 0.5 gives an effort of 50 N, a quarter of the load, for a mechanical advantage of 4. If you instead knew the effort and wanted the load arm, the same equation rearranges to load arm equals effort times effort arm over load. Choosing which term to solve for in the menu simply picks which rearrangement the tool applies.