+4 Most archaeologists believe that the ancient Egyptians used very long ramps while building the pyramids. (It's also likely that ancient Egyptians used levers and ramps in combination.) The ramps had to be long because long ramps are shallow, and the stones the Egyptians assembled into the pyramids were too heavy to lift up steep ramps.
The stones in the upper layers of Khufu's pyramid have a mass of about 2000kg each.
The pyramids were primarily built by farmers, laboring in the seasons when there was no farm work to do. Let's estimate that a single worker can exert a force big enough to pick a 60kg stone straight up. (This means 60kg can play the role of "mass B" from our analysis of a ramp.)
What is the steepest slope for the ramp the ancient Egyptians could have used if each block was to be pushed (and pulled with ropes) by a team of 8 workers?
You can neglect friction for this problem. Although there was friction between the ramp and stone blocks during construction of the pyramids, ancient Egyptians put the stones on sleds, likely lubricated with water, to reduce the friction as much as possible.
This is an estimation problem, so your answer only needs to be approximately correct
The steepest slope for the ramp the ancient Egyptians could have used is about 0.1
Let's assume that the length of the ramp is L meters and the height of the ramp is H meters. The steepness of the ramp is then H/L.
The force that each worker can exert on the stone is 60 kg⋅g=600 N, where g is the acceleration due to gravity (9.8 m/s²). The total force that 8 workers can exert on the stone is 8⋅600=4800 N.
The force needed to push the stone up the ramp is F=mg, where m is the mass of the stone (2000 kg) and g is the acceleration due to gravity (9.8 m/s²).
The steepest slope for the ramp is when the force needed to push the stone up the ramp is equal to the force that 8 workers can exert on the stone. Solving F=4800 N=mg for H/L gives H/L=1/10=0.1.
