Inclined Plane

An inclined plane is the fanciest possible name for a slope: a flat surface tilted at an angle. Pushing or rolling a heavy object up a slope is much easier than lifting it straight up, even though you end up at the same height. The trade-off: you have to travel further. The total work done is the same, but the force you need at any moment is much less. The inclined plane is one of the six simple machines, and possibly the most familiar: ramps, stairs, hills, slides and chutes are all inclined planes.

  • What it isA flat slopeTilted at an angle
  • Trade-offLess force over longer distanceTotal work the same
  • Steeper slopeShorter path, more forceCloser to lifting straight up
  • Gentler slopeLonger path, less forceEasier but slower
  • Used byAncient pyramid buildersAnd every wheelchair ramp today
  • ExamplesRamps, hills, stairs, slidesAnd tilted truck beds

How it makes work easier

To lift a 100 kg load straight up by 1 metre, you need to pull with 100 kg of force (working against gravity).

If you push the same load up a 5-metre ramp to a height of 1 metre, you only need about 20 kg of force at a time, because most of the loads weight is being supported by the ramp itself.

The total work done is the same: 100 kg x 1 m = 100 kg-m, or 20 kg x 5 m = 100 kg-m. The inclined plane spreads the work over a longer distance, so the effort needed at any one moment is less.

Mechanical advantage of a ramp

The mechanical advantage of an inclined plane is the ratio of the length of the slope to the height it climbs:

  • A 5-metre ramp rising 1 metre has a mechanical advantage of 5. Force needed is 1/5 of the loads weight.
  • A 10-metre ramp rising 1 metre has a mechanical advantage of 10. Force needed is 1/10 of the loads weight.
  • The gentler the slope, the higher the mechanical advantage and the easier the push.
Fact The pyramids of Egypt were almost certainly built with the help of inclined planes. Workers built earth ramps spiralling up the pyramid as it rose, dragging stones up the ramps on wooden sledges. As the pyramid grew, the ramps grew too. When the pyramid was finished, the ramps were dismantled. Some ramps may have stretched for hundreds of metres to keep the slope gentle enough for the workers to manage.

Inclined planes around you

  • Wheelchair ramps: required by law in many countries so that people in wheelchairs can enter buildings, replacing or adding to staircases.
  • Loading ramps: lorries and removal vans use ramps to roll heavy boxes up and down.
  • Stairs: a staircase is a sequence of small inclined planes (the treads slope very slightly forward to drain water).
  • Hills and roads: roads zigzag up steep hills using long, gentle inclines instead of going straight up.
  • Slides and chutes: gravity slides goods, post or playground children down a controlled slope.
  • Ski slopes: snow on a long slope lets you slide down with gravity doing the work.
  • Conveyor belts: often built on a gentle slope to lift goods between levels in factories.
  • Boat ramps: launch boats into water without needing a crane.

Why roads zigzag up steep hills

If you drive in mountainous places, the roads usually zigzag back and forth across the slope. Each zigzag (called a switchback or hairpin) lets the road be much less steep than going straight up. A car can easily climb a 6 per cent slope; very few can climb a 20 per cent slope. The longer route is gentler, safer and easier on the engine.

The same logic applies to mountain hiking trails, railway lines climbing through mountains and even the spiral ramps in multi-storey car parks.

Did you know? The famous Lombard Street in San Francisco has 8 sharp switchbacks in a single block, with grades of 27 per cent on the original straight street. The zigzag design was added in 1922 to make the steepest residential block in the city safe to drive on. Without the inclined-plane principle of trading height for distance, no normal car could climb (or descend) that hill safely.

Friction matters

The math above assumes no friction. In reality, friction between the load and the ramp surface always pushes back on the load, requiring slightly more force than the ideal mechanical advantage predicts.

  • Smooth surfaces (well-polished wood, ice) have low friction; loads slide easily.
  • Rough surfaces (rough concrete, sandpaper, gravel) have high friction; loads need more force.
  • Wheels and rollers reduce friction enormously, which is why heavy loads are usually wheeled up a ramp rather than dragged.

Egyptian pyramid builders are now known to have poured water on sandy ramps to reduce friction when dragging stones on sledges. A wall painting from a tomb shows workers pouring water in front of a sled, and modern experiments have confirmed it can cut the force needed by up to half.

Inclined planes for splitting

Two inclined planes joined back-to-back form a wedge, another of the six simple machines. The wedge uses the same principle: a sharp angle multiplies a small force into a much larger splitting force. The shape of a knife blade, axe head or doorstop is an inclined plane in action.

Try this Take a heavy book and a sturdy ruler. Lift the book straight up to a height equal to the rulers length and notice the force needed. Now lean the ruler against a small box to make a ramp, and slide the same book up the ramp to the same height. The push needed at any moment is much less, even though the book ends up at the same height. Try it with different angles of slope: the gentler the slope, the easier the push, but the further you have to slide it.
Deeper dive: building the modern world with inclined planes

The inclined plane is the unseen workhorse of modern engineering. Almost any time goods are moved up or down between levels, an inclined plane is involved somewhere.

The Panama Canal, opened in 1914, uses a system of locks (essentially water-filled inclined planes) to lift ships up to 26 metres above sea level so they can cross from one ocean to the other. Each lock chamber fills or empties with water, gently floating the ships up or down. Without the locks, ships would have had to be cut a sea-level canal through 26 metres of solid Panamanian rock, a much harder engineering job.

The famous Falkirk Wheel in Scotland is a modern alternative: a rotating boat lift that gently rotates ships through 35 metres in a few minutes, replacing 11 separate locks that used to take a day. The wheel itself is a giant balanced machine, but its purpose is exactly the same as a lock: trading height for time using a different version of the inclined-plane principle.

In factories, escalators and travelators carry millions of people every day. Conveyor belts move parcels through Amazon warehouses. Spiral ramps swirl through multi-storey car parks. Roads climb mountains in zigzags. Even the slope of an aircraft wing uses the inclined-plane principle to deflect air downward and lift the plane.

The simple slope, used by humans since the Stone Age, is still one of the most useful inventions ever made.

For more, see wedge and screw.