Inclined Planes

Part of Simple Machines

How ramps, wedges, and screws use the inclined plane principle to multiply force with practical construction examples.

Why This Matters

The inclined plane is the most ancient and most widely applied simple machine. Every ramp, every staircase, every road through hilly terrain, every axe, knife, wedge, and screw is an application of the inclined plane principle. The pyramid builders used inclined planes (ramps) to raise stones that no other ancient technology could have lifted. Every road through mountainous terrain is an inclined plane laid across the landscape.

The principle is elegant: instead of lifting a weight straight up (requiring force equal to the full weight), you slide it along a longer, gentler slope. Less force is required, but it must be applied over a longer distance. The total work (force × distance) remains the same in theory; in practice, some energy is lost to friction, but the reduction in force can be so great that the friction loss is acceptable.

Understanding inclined planes allows you to design solutions to lifting and moving problems that might otherwise seem intractable. How do you get a 200 kg millstone to the top of a building? Not by lifting it — by rolling it up a ramp. How do you split a log without a saw? With a wedge driven by a hammer. How do you hold two pieces of wood together under enormous force without glue? With a screw.

The Physics

Mechanical advantage of an inclined plane:

MA = Length of slope / Vertical height

Or equivalently:

MA = 1 / sin(slope angle)

Force required to push a load up the slope:

Effort = Weight × (Height / Length) = Weight / MA

This is the theoretical force with no friction. In practice, add friction:

Total effort = Weight × (sin(angle) + coefficient_of_friction × cos(angle))

For a smooth wooden surface, coefficient of friction ≈ 0.3-0.5. For a rolling load on wood, rolling resistance ≈ 0.04-0.08.

Example: Rolling a 300 kg stone up a 5 m ramp to a height of 1 m:

• Theoretical: 300 × 1/5 = 60 kg
• With rolling friction (0.06): 300 × (1/5 + 0.06 × √(1-1/25)) ≈ 60 + 18 = 78 kg
• This is achievable by two strong people (39 kg each)

Without the ramp, lifting 300 kg straight up requires 300 kg of force — impossible by hand.

Construction Ramps

Construction ramps are the most direct application of the inclined plane for heavy lifting.

Design considerations:

1. Slope angle: The shallower the slope, the less force required. But a longer ramp takes more material and labor to build. Practical compromise: 10-15% grade (about 1 m rise per 7-10 m length) for rolling heavy stones.

2. Surface: A smoother surface reduces friction and reduces required effort. Options:

   • Wet clay: slippery when wet, cheap, deteriorates with use
   • Wooden planks: good rolling surface, durable
   • Greased wooden sledge runners: very low friction but need continuous grease

3. Width: Wide enough for the load plus clearance on each side for workers. At least 2× the load width for safety.

Building a construction ramp:

For raising stones to construction height (up to 5 m):

1. Plan the ramp angle (12% grade is a reasonable maximum for rolling stones on wood rollers)
2. Build a timber framework supporting wooden planks as the ramp surface
3. Reinforce side rails to prevent the load from rolling off
4. Grease the planks with tallow before each large stone is moved
5. Use a block and tackle or rope team to pull the load while workers steady it from the sides

Temporary earthen ramp: For very heavy loads (large stone blocks), an earthen ramp against the building is sometimes the best solution:

1. Pile earth against the structure up to the required height
2. Compact the surface and add timber planking
3. Use the ramp to raise the heavy component
4. Remove the earth ramp after the component is set

This is wasteful of labor for the removal but avoids the structural complexity of a freestanding timber ramp.

Wedges

A wedge is a moving inclined plane. Instead of moving the load along a fixed slope, you drive the slope (the wedge) into or under the load.

Applications:

• Log splitting: An axe head is a wedge; the hammer blow drives the wedge through the wood
• Stone splitting: Iron wedges driven into drilled holes split stone along a controlled line
• Lifting: Wedges driven under a heavy load lift it incrementally (combined with lever or crib)
• Tightening: Wedges driven into joints (like timber mortise-and-tenon connections) lock them without fasteners
• Cutting: Every knife, chisel, and drill bit is a wedge in some form

MA of a wedge:

MA = Length of wedge / Width at the thick end

A splitting wedge 20 cm long and 5 cm wide at the thick end: MA = 20/5 = 4:1

The high mechanical advantage plus the inertia of a hammer blow (impact force is much greater than static force) makes wedges capable of developing enormous splitting forces.

Wedge material:

• Iron or steel: required for splitting stone or hardwood in serious quantities
• Hardwood: adequate for light work and wood splitting in very soft timber
• Dried bone: emergency use, splitting soft materials

Splitting stone with wedges: This technique (known since ancient Egypt) allows cutting stone along precise lines without explosives:

1. Drill a series of holes along the desired split line, spaced 10-15 cm apart, 5-8 cm deep
2. Insert iron wedge + two thin iron shims (feathers) into each hole
3. Drive each wedge in sequence, a few hammer blows each, working down the line
4. Repeat, increasing force gradually
5. The stone splits along the line of holes as internal stress builds

Shims (feathers and wedge): The two thin iron shims on each side of the wedge prevent the wedge from sticking in the hole and allow the wedge force to be transmitted as outward splitting force perpendicular to the wedge direction.

Screws as Inclined Planes

A screw is an inclined plane wrapped around a cylinder. Each turn of the screw advances it by one pitch — the same as moving a load along the equivalent length of inclined plane.

The screw’s unique advantage: Extremely high mechanical advantage, combined with the self-locking property. Unlike most other simple machines, a screw under load does not run backward when you release the driving force — the thread friction holds the position. This makes screws ideal for clamps, vises, and presses where the force must be maintained without continuous operator input.

MA of a screw:

MA = (2π × lever radius) / thread pitch

This is always very large for practical screws. A screw with a 2 cm pitch and a 40 cm lever radius: MA = 2π × 40/2 = 125:1

Cutting threads without a lathe:

1. A master screw (a piece with threads already cut, acting as a template) can cut threads by being driven into a split nut blank (a nut made from two halves that can be opened to insert the master screw and closed to cut threads)
2. Alternatively, file a thread profile by hand: mark the helix on the outside of a cylinder, then file the groove along the marked line
3. The mating nut is cut using the threaded rod as the tap

Types of screw thread for hand manufacture:

• Square thread: easier to cut, stronger for power applications (presses, vises)
• V-thread (triangular): slightly more self-locking, commonly used for fasteners
• Buttress thread (one face steep, one shallow): specialized for one-direction force applications

Ramps for Vehicle Loading

One of the most common inclined plane applications is loading vehicles.

Loading dock ramp: A permanent or semi-permanent ramp from ground level to wagon bed height, used to roll barrels, push wheelbarrows, or slide sacks without heavy lifting.

Dimensions for a standard farm wagon (bed height 80-90 cm):

• Ramp length: 4-5 m (for a 12-15% grade)
• Width: 1.0-1.2 m
• Surface: Planks with cleats (horizontal strips) every 30-40 cm to prevent sliding backward

Boat ramp (slipway): For launching and retrieving boats:

1. The ramp slope typically 10-15% below water to above high tide mark
2. Greased timber runners reduce friction to allow even large vessels to slide
3. Capstan or block-and-tackle provides the pulling/lowering force

Switchback Roads as Applied Inclined Planes

A switchback road is an inclined plane traversed diagonally (or with direction reversals) to achieve a tolerable gradient over steep terrain.

Design:

• Each switchback section is a separate short inclined plane
• The grade on each section: typically 6-8% for loaded vehicle traffic
• The switchback corners: wide enough to turn a vehicle (8-10 m radius minimum for horse-drawn wagons)
• Retaining walls on the outer edge of each corner are critical structural elements

The overall rise is achieved by the sum of height gains on each switchback section. A hill of 100 m height with 8% maximum grade requires at least 100/0.08 = 1,250 m of total road length to cross it.