"Sled on ramps" is the most favoured transport method of the egyptologists. Unfortunately most people are unable to see the difference between lifting a stone 1 m, or pulling that stone up on a ramp. That, for example, the weight of a block is relatively irrelevant for the ramp model. To understand this some physics is necessary. But don't despair, I have explained it as simple as it could be.

What is "force"?

Everybody knows the term, nobody can explain it. In the ideal world of friction free physics force (F) is something you have to deliver to accelerate an object in a given time to a certains speed. In the real world you also must deliver force to push an object around, or to hold an object above the ground. Everything understood?
Surely not. The physical definition of force is F=m*A or mass multiplied with acceleration. Force is measured with the unit Newton (N) which has the definition 1 N = 1 kg * 1m/s2. If you accelerate one kilogram in one second to the speed of 1 meter per second you have used the force of one N(ewton).
Our Earth is a mass magnet which tries to pull down any object into the direction of its centre with an acceleration of about 10m/s2. To hold a weight of 1 kg in the air you therefore have to deliver a force of 10 N to it, in a direction opposite to the weight force of the earth. Or, to see it the other way round, 1 kg has the weight force of about 10 N.
In the ideal, friction free world most physicists love no force is needed to keep a body in uniform motion perpendicular to any force (eg. horizontally ofer a friction free table). The body moves until he falls down or is stopped by using a counter force. The real world is not so nice: we have "friction". If we move an object around on a table the million microscopic irregularities of the surfaces of desk and object interlock with each other, pulling on table and object and decelerating the object. Because deceleration is nothing else but an acceleration, this friction can be called a force, too. It's the friction force Ff.
The last thing we have to know is the principle of "Actio = Reactio" which was discovered py the great English physicist Isaac Newton. It's the basis for rocket fligth and ramp transport. It says that any force delivered to a static body produces a counter force of the same strength but in opposite direction. Or: to move a body which is hold by a force you have to use a greater force in the opposing direction.

The ramp

To lift a pyramid block of 2500 kg you need a force greater than its weight force of 25000 N. To pull the same block up a ramp, you have the additional friction force to come over. So what kind of force you need then? Friction force AND weight force? Unfortunately this is the opinion of many people I asked. But it's wrong. Fortunately.

inclined plane A ramp works like this inclined plane. The weight force of the block Fg is split into two seperate forces perpendicular to each other. The "normal force" Fn which presses the block down and is important for the friction force, and into the sliding force Fh which pulls the block down parallel to the surface of the inclined plane.

The steeper tha ramp the larger the amount of the sliding force, when the "ramp" is vertically (and Alpha = 90) only the sliding component is left and equals the weight force. On the other extreme (the ramp is horizontal) the sliding force vanishes and only the component of the normal force is left.

Therefore the sliding force is defined as Fh = Fg * sin(Alpha)
and the normal force Fn = Fg * cos(Alpha).
In the friction free environment of the physicists the sliding force of a 2.5 t stone on a 5 ramp (about 2 fingers to the cubit) is about 2170 N! less then 1/10th of the weight force!
But we have to take friction in account. The friction depends only on the measureable friction coefficient and the normal force. The friction coefficient for wood on gravel is between 0.18 and 0.3 dependent of the gravel size, I use a value of 0.25 for the next calculations.
Therefore the friction force on a ramp can be calculated to Fr = Fn * r, and is about 6220 N for a 2.5 t stone. Friction force and sliding force together are around 8400 N, about 1/3rd of the weight force.
Of course there are 2 friction forces to be taken into calculation: the sliding friction which works on a moving body, trying to decelerate it, and the static friction which works on the resting body, opposing it to move. The static friction is larger than the sliding friction, which is why for exampe resting sleds need a large push to start, and only gentle force while in motion.


Perpetuum Mobile "Just a moment" some will interject now. "If I need only 1/3 of the weight force to move a stone up, I can build a perpetuum mobile by pulling 2 1/2 stones up with the force used by one stone going down". The end of all energy problems?
Unfortunentely no. Our 5-ramp is about 12 m long for each meter in height. Sure I can use the weight force of an 850 kg block to pull up a 2500 kg block - but only 1 m up the ramp. I would have to use 12 blocks with 850 kg each to pull the new block up the whole ramp. And even in a friction free world have to need 12 blocks of 217 kg each to pull the new block up. Both processes need the same amount of work
"Work" is the sum of all forces and all ways used to do a task: "Force X distance". Work is measured with the unit "Joule". To move the block 1 m up the friction free ramp you would use (2170N x 12 m) = 26040 J. To pull up 12 times the counterweigth of 217 kg I need 26040 J, too!
The friction in the real world makes matters worse. Do I need 25000 J to lift a block up 1 m, I need four times the work to pull it up a slope.
But then a ramp does make no sense! Wrong. We humans have the possibility to spend large amounts of work during the day, but our force is limited. All machines invented in the past were used to reduce the needed FORCE although every invention enlarged the amount of work needed. Blocks and tackles have inner friction which adds work to the process, the same with levers and pulleys. But FORCE is the critical parameter.

Lift, carry and pull

Lets come to the last point: How much force can be asserted to a rope from a worker? Some workers can carry 50 kg (500 N) for some meters, but I don't think that this is possible the whole day long. on the other hand it is totally easy to pull with a force of 500 N for a long time - try it out in a sport studio!
The reason: The force comes not from the weaker arms, but from the stronge legs. And from the GRAVITY! To pull a block you can use the same force which tries to pull the block down the ramp to pull it up, when you bend your body forwards.
The effect: it is possible to pull even with 1000 N for a longer time, if we take half of it, 500 N, we need about (8400 N / 500 N) = 16.8 worker for a 2.5 t stone up a ramp.

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All pictures and texts © Frank Dörnenburg