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Homework Help: Basic mass and density problem

  1. Sep 3, 2009 #1
    This is one of my homework problems for my mechanical engineering class. The problem is extremely simple, but, the homework is graded in this class and I want a good grade :). I should be able to do this no problem but I am getting confused by what some of the reading in my book is telling me.

    1. The problem statement, all variables and given/known data
    As shown in given figure, a cylinder of compacted scrap metal measuring 2m in length and 0.5m in diameter is suspended from a spring scale at a location where the acceleration of gravity is 9.78 m/s2. If the scrap metal density in kg/m3, varies with position z, in m, according to p = 7800-360(z/L)2, determine the reading of the scale in Newtons.

    Cylinder diameter=.5m
    Cylinder height=2m
    http://img30.imageshack.us/img30/3094/0903091713.jpg [Broken]
    2. Relevant equations
    Volume of Cylinder= 3.14r2*H
    Density of cylinder=7800-360(z/L)2

    3. The attempt at a solution
    I already found the volume which is .3925m3. What I would normally do is solve for mass using the d=m/v equation. But my book says mass=[tex]\int[/tex](p)dV. It also says that "density, p, at a point is defined as, p = lim(from v to v')m/V". I have taken physics and lots of math but using limits and integrals for finding masses and densities is throwing me off.
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 3, 2009 #2
    The problem is that the density is not constant. Therefore, you would be well-advised to do the triple integral.
  4. Sep 3, 2009 #3
    Hmm it makes sense that the density is not constant. But I dont understand how I am supposed to solve that integral.
    If mass=[tex]\int[/tex](p)dV, and p = 7800-360(z/L)2, "z" can be any number between 0 and 2. Also the integration is with respect to V, which is not in the given density formula. Thanks for the reply.
  5. Sep 3, 2009 #4
    Hint: Since density only varies along the length of the cylinder (and not in any other directions) you can reduce the triple integral mentioned earlier to a single integral.

    If the mass is cylindrical, then what is the formula for its volume?

    Also, how does z relate to the variables used in determining the volume of the cylinder?
  6. Sep 4, 2009 #5


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    Another hint:

    What is the mass of a slab of thickness dz? Assume the density does not vary significantly within the slab.
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