أنشئ حسابًا أو سجّل الدخول للانضمام إلى مجتمعك المهني.
Quantity of materials required for different works?
The definition for the second moment of inertia IcIc for a filled and hollow cylinder can be found on http://en.wikipedia.org/wiki/Second_moment_of_area: Ic=∫∫Ay2dxdy=∫R0∫2π0r2dϕ rdr=πr44Ic=∫∫Ay2dxdy=∫0R∫02πr2dϕ rdr=πr44
The surface area of the filled cylinder is:
A=πr2A=πr2
Compare filled and hollow cylinder of equal mass:
sc=IcA=r24sc=IcA=r24, cylinder with fractional internal radius ri=xrori=xro and x<1x<1:
sh=IhA=r4(1−x4)4r2(1−x2)=r2(1−x2)(1+x2)4(1−x2)=r2(1+x2)4>scsh=IhA=r4(1−x4)4r2(1−x2)=r2(1−x2)(1+x2)4(1−x2)=r2(1+x2)4>sc.
This means a hollow cylinder is stronger than a rod of equal mass and the same material. A hollow cylinder with a bigger inside diameter is better. In the limit x→1x→1 the hollow cylinder is twice as strong. Note that this limit isn't physically viable as it would be an cylinder with infinite radius and infinitesimally thin wall. However it is useful to define the upper limit of the second moment of inertia. I didn't expect the increase in strength only a factor of two.
