Inclined Planes, Wedges, and Screws
An inclined plane is a sloping surface that lets a load be raised with less force over a longer distance, a wedge is a moving inclined plane that splits or lifts material, and a screw is an inclined plane wrapped around a cylinder; their ideal mechanical advantages are length ÷ height, length ÷ thickness, and circumference ÷ pitch. In brief: for a frictionless ramp of length L and height h, the effort force needed to push a load of weight W up is F = W × h ÷ L, so IMA = L ÷ h. A gentler slope means less force, but a longer distance, and the work done against gravity remains the same.
Ramps, knives, axes, nails, bolts and jar lids all use the inclined plane principle. This guide explains how the three machines work, compares them in a table, works through calculations for a ramp with friction and a screw jack, and corrects common mistakes.
How does an inclined plane work?
To lift a load straight up through a height h, a force equal to its weight W is needed. If the load is pushed up a ramp of length L, the work against gravity is the same, W × h, but this is done over a distance L, so the force along the ramp is smaller: F = Wh ÷ L (without friction), which is W sin θ for a slope of angle θ. The ideal mechanical advantage is IMA = L ÷ h = 1 ÷ sin θ. With friction, an additional force f = μN opposes the motion, so F = W sin θ + μW cos θ for a ramp, and the efficiency is lower. Ramps for wheelchairs are made with gentle slopes (about 1:12) so that the force required is small.
How does a wedge work?
A wedge is two inclined planes joined back to back, forming a thin end and a thick end. When it is driven in by a force on its blunt end, it exerts large forces at right angles to its sloping sides, which push material apart (axe, chisel, knife, nail, a splitting wedge) or lift a load (a door stop, a wedge under a heavy machine). The ideal mechanical advantage is the length of the slope divided by the thickness of the wedge: IMA = L ÷ t. A thinner wedge has a greater advantage, which is why a sharp knife cuts better than a blunt one.
How does a screw work?
A screw is an inclined plane wound around a cylinder as a helical thread. The distance between neighboring turns of the thread is the pitch p. One turn of the screw advances it by one pitch. For a screw turned by a handle or a screwdriver of radius r, the effort moves around a circle of circumference 2πr during one turn, while the load advances by p, so IMA = 2πr ÷ p. A fine thread (small pitch) gives a large advantage but requires many turns. Applications: bolts, wood screws, jar lids, car jacks, vises and the propeller (Archimedes’ screw for lifting water).
The three machines compared
| Machine | Ideal MA | Is the object or the machine moved? | Examples |
|---|---|---|---|
| Inclined plane | L ÷ h | The load moves along the plane | Ramp, mountain road |
| Wedge | L ÷ t | The wedge moves into the material | Axe, knife, nail |
| Screw | 2πr ÷ p | The screw turns and advances | Bolt, jack, jar lid |
Worked example: a ramp and a screw
- Ramp, frictionless. A 200 kg crate is raised onto a platform 1.0 m high using a 4.0 m ramp. W = 200 × 9.8 = 1,960 N. F = W × h ÷ L = 1,960 × 1.0 ÷ 4.0 = 490 N. IMA = 4.0.
- Ramp with friction. If the friction force is 100 N, the effort is 490 + 100 = 590 N. The actual MA = 1,960 ÷ 590 = 3.3, and the efficiency = MA ÷ IMA = 3.3 ÷ 4.0 = 83 percent. Check with work: output 1,960 × 1.0 = 1,960 J; input 590 × 4.0 = 2,360 J; 1,960 ÷ 2,360 = 83 percent.
- Screw jack. The thread has a pitch of 2.0 mm, and the handle has a radius of 0.10 m. IMA = 2π × 0.10 ÷ 0.0020 = 314. With an efficiency of 30 percent (typical for jacks because of high friction), the actual MA = 0.30 × 314 = 94, and a force of 100 N on the handle can lift a load of 9,400 N, about 960 kg.
- Wedge. A wedge 10 cm long and 2.0 cm thick has IMA = 10 ÷ 2.0 = 5, so an ideal driving force of 500 N produces a splitting force of 2,500 N.
How were they developed?
The ramp was used in building the Egyptian pyramids to raise stone blocks, and wedges are among the oldest tools (stone axes, over a million years old). The screw was described by Archimedes as a water-lifting device, and the screw press was used by the Romans for olive and wine. Henry Maudslay built an accurate screw-cutting lathe in 1800, making standardized screws possible.
Which mistakes are common?
Mistake 1: Using the horizontal length of the ramp
Use the length along the slope for L, and the vertical rise for h.
Mistake 2: Thinking that a ramp reduces the work
It reduces the force, but the work is the same or larger because of friction.
Mistake 3: Confusing pitch with the number of threads per unit length
Pitch is the distance between threads (mm per turn), the inverse of threads per mm.
Mistake 4: Forgetting that screws are inefficient
Friction in the thread is large. This is useful, because it holds the load without slipping back (self-locking).