Inclined Plane, Screw & Wedge

the same triangle, wrapped round a cylinder or driven into a log

Worked example — Inclined Plane, Screw & Wedge

Ideal MA	2.00
Effort Force (E)	336.6 N
Actual MA (with friction)	1.49
Efficiency (η)	74.3%
Ramp Angle (θ)	30°
Screw Pitch (p)	5.0 mm
Shaft Radius (r)	10 mm
Wedge Angle (α)	20°

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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A gentler ramp always wins, and always costs more to win by 🖖

Ideal advantage is 1/sin θ, so shallower is better on paper: at 15° the geometry offers 3.86 against 1.41 at 45°. Friction takes most of that back. Hold μ at 0.2 and what you actually get is 2.21 against 1.18, so going gentle buys 1.03 of real advantage where the geometry promised 2.45 — 42% of it. Sweeping every angle shows no turning point: 5° is better still at 3.49, and better again below that. What you pay is efficiency, down to 30.4%, and 11.47 metres of pushing for every metre of height.

A screw is this ramp rolled up 🖖

Unwrap one turn of a thread and you have a triangle: rise equal to the pitch, base equal to the circumference. The standard screw preset has a 3 mm pitch on a 10 mm radius, so its slope is 2.734° and its ideal advantage is 20.94 — a number no ramp you could walk up will give you. The price is on the same line: at 24.0% efficiency, three quarters of the work goes into the thread. Screws are for force, never for moving things.

Below one angle it stays where you put it 🖖

A load holds on a slope when tan θ is no greater than μ, and slides back the moment it is not. At μ = 0.2 the boundary sits at 11.31°: at 11.3° the load holds, at 11.4° it goes. This is the same condition that makes a screw self-locking, and it is why a bolt stays done up with no ratchet and no pawl. A machine that is bad at giving work back is a machine that does not need a brake.

Forty times the weight, and not one number moves 🖖

Drag load from 50 N to 2000 N and watch the verdict above the ramp. It does not change. Neither does the advantage: at 20° with μ = 0.36 the tool reads 1.4699 for a 50 N crate and 1.4699 for a 2000 N one. Weight enters the problem twice and divides out. It pulls the load down the slope as mg sin θ, and it presses the load into the ramp as mg cos θ, which is the force friction has to work with. Double the mass and you double both, so the holding condition is tan θ ≤ μ and there is no m anywhere in it. A pebble and a piano let go on the same slope at the same instant stay level all the way down. The effort does scale, of course: 34.02 N against 1360.62 N, exactly forty times as much. The ramp offers every load the same bargain and has no opinion about how heavy it is.

Problem solved in full

Unrolling one turn of a screw thread into a triangle 7 steps
Load Standard Screw: a 3 mm pitch on a 10 mm radius, 1,000 N of load, μ = 0.15. The panel says an ideal advantage of 20.94 and an efficiency of 24.0%. Both come from the same triangle. Find it.
1. Take one full turn of the thread and unroll it. The thread climbs by the pitch while it travels once round the shaft, so it is the hypotenuse of a right triangle whose height is the pitch and whose base is the circumference. A screw is an inclined plane that has been wrapped round a cylinder, and this is that sentence made arithmetic.
  
  $one turn unrolled: rise p = 3 mm, base 2 π r = 62.83 mm$
2. The slope of that triangle is the helix angle. Under three degrees, which is why a screw feels nothing like a ramp even though it is one.
  
  $λ = arctan \frac{p}{2 π r} = arctan \frac{3}{62.83} = 2.73 4^{\circ}$
3. Ideal advantage is the distance you move over the distance the load moves: once round the circumference to advance one pitch.
  
  $MA_{ideal} = \frac{2 π r}{p} = \frac{62.83}{3} = 20.94$
4. Friction enters as an angle of its own. The friction angle is the slope at which a surface just begins to let go, and adding it to the helix angle is the standard way to fold friction into a screw.
  
  $φ = arctan μ = arctan 0.15 = 8.53 1^{\circ}$
5. The effort follows directly. Rather than 1,000 divided by 20.94, which is 47.8 N, you need four times that.
  
  $E = L tan (λ + φ) = 1000 tan (11.26 5^{\circ}) = 199.2 N$
6. The advantage you actually get is the load divided by that effort, 1,000/199.2 = 5.02. So the efficiency is the ratio of what you got to what the geometry offered.
  
  $η = \frac{MA _{actual}}{MA _{ideal}} = \frac{5.02}{20.94} = 24.0%$
7. And the same comparison, written with a tangent instead of an angle, tells you whether the screw stays put with no hand on it.
  
  $tan λ = 0.0477 \leq μ = 0.15 \Rightarrow self-locking$
Answer

An ideal advantage of 20.94 from a triangle 3 mm tall and 62.83 mm long, and 24.0% efficiency because friction adds 8.531° to a slope that was only 2.734° to begin with.

Three quarters of the work goes into the thread, and that is not a defect being tolerated — it is the feature being paid for. A machine this bad at giving work back cannot be pushed backwards by its own load, which is precisely what tan λ ≤ μ says: 0.0477 against 0.15, so it self-locks with room to spare. That is why a bolt stays done up with no ratchet, no pawl and no spring, and why a car jack holds a ton in the air while you take your hand off the handle. Switch the mode to Ramp and try to find an angle that self-locks and still gives an advantage of 20: there is none, because on an open ramp the two conditions pull in opposite directions. Wrapping the triangle round a cylinder is what lets a screw have both.

Learning path

Simple machines: trading force for distance

Leads to Gear trains friction as an angle rather than a percentage, and with it self-locking: the reason a bolt stays done up.

By the 42Math team · Updated 2026-08-31

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Gentle Ramp (15°) - A 15° ramp with μ = 0.2: 500 N pushed with 226.0 N. The geometry offers 3.86 and friction returns 2.21, at 57.3% efficiency.
Steep Ramp (45°) - A 45° ramp with μ = 0.3: 459.6 N of push for barely any advantage at all, 1.09 — but the most efficient setting here, at 76.9%.
Standard Screw - A 3 mm pitch on a 10 mm radius: a 2.734° slope, an ideal advantage of 20.94, and 24.0% efficiency. Self-locking, as every bolt must be.
Splitting Wedge (15°) - A 15° splitting wedge at μ = 0.25: 631.4 N of hammer to deliver 800 N sideways, and 16.7% efficiency. The least efficient machine in the chapter, and it still splits the log.