A sledge carrying a stone block is hauled up a ramp of angle θ at (near) constant velocity. Resolving forces along the slope, the pulling force needed to overcome gravity's slope component and Amontons–Coulomb kinetic friction is:
F_required = μ·m·g·cosθ + m·g·sinθ
where m is the block+sledge mass and μ is the sand's kinetic friction coefficient. Dry desert sand gives μ ≈ 0.55 for a wood sledge runner. A 2014 physics study (Fall, Weinhart, Bonn et al., Amsterdam) measured that wetting sand ahead of the sledge with the right amount of water pulls the grains together by capillary bridging, roughly halving the drag — μ ≈ 0.20 — matching a wall painting at Djehutihotep's tomb that shows a man pouring water in front of a sledge hauling a colossal statue.
Each hauler contributes a documented sustainable pulling force of about 250 N (peak pulls run higher but cannot be held for a multi-metre haul), reduced by a 0.9 rope-team efficiency factor for imperfect rope angles among many haulers:
F_available = N_workers · 250 N · 0.9
Moves if F_available > F_required
N_needed = ceil(F_required / (250 N · 0.9))
When F_available exceeds F_required, the excess force accelerates the sledge: a = (F_available − F_required) / m, capped to a realistic slow walking haul speed.
- Wet sand toggle halves μ, dropping the force (and worker count) needed — try it on a steep ramp with a heavy block.
- Ramp angle beyond ~10–15° needs disproportionately more haulers even on wet sand, which is why real Giza-era ramps used shallow, long slopes.
- Block mass up to 80 t models the largest granite blocks moved in antiquity (e.g. the King's Chamber roof beams).