How Do You Calculate the Scale Reading for a Varying Density Cylinder?

[stevebrstlct]

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.

Homework Statement

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
G=9.78m/s²
http://img30.imageshack.us/img30/3094/0903091713.jpg

Homework Equations

F=mg
Volume of Cylinder= 3.14r²*H
Density=M/V
Density of cylinder=7800-360(z/L)²

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=\int(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.

 

[planck42]

The problem is that the density is not constant. Therefore, you would be well-advised to do the triple integral.

 

[stevebrstlct]

Hmm it makes sense that the density is not constant. But I don't understand how I am supposed to solve that integral.
If mass=\int(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.

 

[PhaseShifter]

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?

 

[Redbelly98]

Another hint:

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