Moving a stone: friction and wet sand

The pyramid builders had no engines. For most of the job the heavy lifting meant one thing: pulling stone on a sledge. How hard is that? You can work it out with the friction model from the Forces & Motion lab, and the answer shows a clever trick the Egyptians seem to have used.

Flat ground. To start a block sliding, the pull must beat friction:

F = μ × m × g

where μ (mu) is the friction coefficient, m is the mass and g ≈ 9.81 m/s². Take a block of 2.5 t (2,500 kg). Its weight is mg ≈ 24.5 kN (kN means thousand newtons). On dry sand a sledge has μ ≈ 0.5, so

F ≈ 0.5 × 24.5 kN ≈ 12.3 kN.

Uphill. On a slope of angle θ, you must also lift the block, so

F = m g (sin θ + μ cos θ).

For a slope of 1 in 10 (θ ≈ 5.7°), sin θ ≈ 0.0995 and cos θ ≈ 0.995. With μ = 0.5, F ≈ 24.5 × (0.0995 + 0.4975) ≈ 14.6 kN.

The wet-sand trick. In 2014 a team of physicists (Fall and colleagues, published in Physical Review Letters) measured sledges pulled over sand in the lab. When the sand is dry, the sledge pushes a heap of sand ahead of it, and the heap makes it hard to pull. When the sand has just the right amount of water, it becomes stiffer, the heap does not build up and the sledge slides. In their experiments the pulling force dropped to about half, so the effective coefficient fell from μ ≈ 0.5 to about 0.25. Too much water makes a slurry and the benefit fades. (Coefficients depend on the sand and the sledge, so these are typical values, not constants of nature.)

With μ = 0.25, the same 2.5 t block needs F ≈ 0.25 × 24.5 ≈ 6.1 kN on the flat, and F ≈ 24.5 × (0.0995 + 0.2488) ≈ 8.5 kN up the 1 in 10 slope.

An ancient picture. A tomb painting from about 1900 BCE, in the tomb of the official Djehutihotep, shows a large statue being dragged on a sledge by many men. In front of the sledge a man pours liquid onto the ground. For a long time this was read as a ritual act. The physics shows that wetting the sand in front of a sledge really does make pulling easier, so it may have been practical as well. That is an inference: the picture does not say why the man pours.

How many people? We need an assumption for one person's sustained pull. We use 0.4 kN (about 40 kgf) as a round teaching value. Real sustained pulls over a long day are uncertain and may well be lower, in which case the teams must be bigger.

• Flat, dry sand: 12.3 ÷ 0.4 ≈ 31 people
• Flat, wet sand: 6.1 ÷ 0.4 ≈ 15 people
• 1 in 10 slope, dry sand: 14.6 ÷ 0.4 ≈ 37 people
• 1 in 10 slope, wet sand: 8.5 ÷ 0.4 ≈ 21 people

Wet sand roughly halves the team on the flat and cuts it by about 40 percent on the slope.