How can the weight of the cone influence the work done?

In summary, the problem involves a conical mound of height h built by slaves to commemorate a victory. The total weight of the finished mound is M and the slaves are instructed to heap up uniform material found at ground level. The work done by the slaves is equal to (1/4)hM, as shown by the equation dW = dF(distance) and the integral W = ∫ρ*dV*(distance)*dx. The density ρ is equal to 3M/(\pi R^2h) and the volume of each layer of the cone can be represented by \pi(R^2/h^2)z^2dz, where dz is the thickness of each layer. The total work can be found
  • #1
Sarangalex
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Homework Statement



A great conical mound of height h is built by the slaves of an oriental moarch, to commemorate a victory over the barbarians. If the slaves simply heap up uniform material found at ground level, and if the total weight of the finished mound is M, show that the work they do is (1/4)hM.


Homework Equations



dW = dF(distance)
W = ∫ρ*dV*(distance)*dx

The Attempt at a Solution



I said dF is equal to ρ*dV and the distance is x.
dV should be equal to ∏r2h*dx.

I just really don't know what to do from this point. What does the given M have to do with anything?
 
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  • #2
"dx" is not "distance moved", it is the height to which a particular "piece" (dV) of Earth has to be raised. You are told that M is the mass of the finished cone so M is the integral of [itex]\rho dV[/itex]. dV cannot be "[itex]\pi r^2 h dx[/itex]", that has the units of distance4, not distance3. The full volume of a cone of height h and radius R is [itex](1/3)\pi R^2h[/itex]. Since that has weight M, your density is [itex]\rho= 3M/(\pi R^2h[/itex].

If the final cone has height h and base radius R, then the radius of the cone at height z is r= (R/h)z so each cross section would be a disk of are [itex]\pi((R/h)z)^2[/itex] and you can "build" the cone out of disks of volume [itex]\pi(R^2/h^2)z^2dz[/itex] where dz is the "thickness of each cone". Multiply that by [itex]\rho[/itex] to get the weigth of that "layer" of Earth and by z for the heigth to which it was lifted. That gives the work done in lifting that particular "layer" of earth. Integrate to find the total work.
 

What is a conical mound?

A conical mound is a type of man-made structure that resembles a cone or pyramid in shape. It is typically constructed by piling up earth, rocks, or other materials in a circular or conical shape.

What is the purpose of building a conical mound?

The purpose of building a conical mound varies depending on the culture and time period in which it was constructed. Some conical mounds were used as burial sites, while others served as platforms for important buildings or as markers for ceremonial or religious purposes.

How were conical mounds built?

The construction of conical mounds typically involved the use of manual labor and simple tools such as shovels and baskets. The process would begin with clearing the area and leveling the ground, followed by piling up layers of earth or other materials until the desired height and shape were achieved.

Where can you find conical mounds?

Conical mounds can be found all over the world, with some of the most well-known examples located in North America, Europe, and Asia. They can be found in a variety of environments, including flat plains, river valleys, and hilltops.

What can we learn from studying conical mounds?

Studying conical mounds can provide insights into the beliefs, practices, and daily lives of the cultures that built them. They can also offer valuable information about ancient construction techniques, land use patterns, and societal organization.

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