Screw Jack Mechanical Advantage

A screw is a wrapped ramp. Unwrap one turn of the thread and you get an inclined plane of length sqrt((pi d)^2 + L^2) and height L, where L is the lead (advance per turn) and d the mean thread diameter. The handle trades distance for force: with no friction, effort times the path of your hand 2 pi l equals the load times the lead, so the ideal mechanical advantage is 2 pi l / L. A fine thread (small L) and a long handle give a huge advantage, at the price of many turns.

Friction. With a thread friction coefficient mu on the ramp, the torque to raise the load is T = (W d/2)(tan(lam) + mu)/(1 - mu tan(lam)) plus the friction of the thrust collar, W mu_c d_c / 2. The efficiency e = W L/(2 pi T) is the useful work over the work you do. When mu is at least tan(lam) the lowering torque is positive: friction beats the load's pull and the screw is self-locking, which is why jacks and vises stay where you leave them (and why efficient screws such as ball screws are not). A self-locking screw cannot be more than 50 % efficient.

What is simplified. Square threads (an Acme or metric thread has a flank angle that raises the effective friction), constant friction coefficients, no starting (static) friction peak, no elastic deformation of the screw, no inertia of the handle, perfect axial load. The jack is drawn at its real size; the load block is only a visual marker, the load is the mass slider.

Numerical checks (run on load): the closed-form torque against a numerical solution of the force balance on the unwrapped thread; the energy over one turn split into load energy, thread friction and collar friction; the maximum efficiency found by scanning the lead angle against (1 - sin phi)/(1 + sin phi); and the lead of the three Blender-built threads measured from their mesh vertices, together with the rendered rise of the platform per turn.

• Turn the handle with the default settings, then switch to Rusty and dry: the effort grows and the lowering torque turns positive, so the jack self-locks.
• Pick the 3 mm fine lead: the mechanical advantage doubles compared with 6 mm because the load rises half as far per turn.
• Press Frictionless (ideal) and compare the energy bar: all the hand work becomes load energy.