Assignment: An ancient engineering mystery / Mechanics & Hydraulics
The big pyramid of Cheops was built from 2551 BC till 2528 BC. That is in a time period of 23 years. See also lecture 3.1. There exist different hypothesis on how the pyramids were constructed. In this assignment you will investigate one of these hypothesis where the ancient Egyptians made use of big water pipes, along which the stones were transported using buoyancy (figure 1).
Figure 1. The stones were surrounded by air bags made of animal skin, so that they float on water (a). The wrapped stones were then transported to the foot of the pyramid where they were guided into a water pipe that was constructed onto the pyramid (with inclination 51.8 degrees) (b). As a result of the upward buoyance force, the stone carrier moves upward in the pipe until it reaches a first sluice gate. At that moment, a similar sluice gate at the foot of the pyramid is closed and the gate that was reached by the stone is opened. The sequence is repeated until the stone reaches its destination (c). The dimensions of stone with carrier bags and the water pipe are shown in (c).
The sluice gates are opened one by one when the wrapped block has stopped by the gate. A block starting in O travels upwards with gate A closed. When the block arrives at gate A, it is stopped by the closed gate. Then the gate at O is closed and gate A is opened, with gate B closed. While the block is travelling from gate A to gate B, gate A is closed. The block arrives at block B that is still closed. Then gate B is opened. Etc. The time to open a gate is 10 seconds.
The blocks with carrier air bags experience friction (as indicated in figure 1c). Assume that the stone carrier is a sphere with the dimensions indicated in figure 1c. One stone measures 0.7 m x 1.1 m x 1.3 m and weighs 2500 kg.
Questions:
1. What is the time needed for the block to travel one column? Assume that after the door is closed some air heaps up at each gate. Assume that opening and closing the gate takes approximately 10 seconds.
2. What is the travel time for one block all the way up to the top of the pyramid?
The Babylonians used water clocks for time keeping (figure 2a). A conical recipient (left) is filled with water. A circular hole (diameter = 2 mm) is drilled at a depth of 30 cm.
Figure 2. Babylonian water clock (a) and dimensions (b). A theoretical water clock with a certain recipient shape for which the change in water level is constant (c).
3. After how much time has the level lowered with 20 cm (i.e. 10 cm above the hole)?
4. What does the shape of a recipient has to look like in order to have the level h decreasing at a constant speed of 2 mm/s (mathematical function)?
The Greek used algebra to solve geometrical problems such as the intersection of cones. Some related geometric problems are given below.
5. A cone with a conical angle of 60 degrees is intersected by a cylinder with radius 1 m. What is the cross-sectional volume bounded by the cylinder, the cone and the plane z = 5 m as shown in figure 3 (subscribed by the red line).
6. Verify by using Matlab that the calculated value (in exercise 5) is correct. (Provide the Matlab code)
7. A parabola with equation y = x2 is intersected with a line as indicated from points A(-1,1) to B(2,4) as illustrated in figure 4. What are the coordinates of the point C for which the surface area S of the triangle described by ABC is maximal?
Hint: note that the surface of a triangle can be expressed as a vector product.
Figure 4. Triangle subscribed in a parabola.