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

In summary, the student is getting confused by what they are reading in their book and is looking for help from others in the class. The student has found the volume of the cylinder and is trying to solve for mass. However, they are getting confused by the limits and integrals used in solving for mass and density.
  • #1
stevebrstlct
4
0
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=[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.
 
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  • #2
The problem is that the density is not constant. Therefore, you would be well-advised to do the triple integral.
 
  • #3
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=[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.
 
  • #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?
 
  • #5
Another hint:

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

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

What is mass and how is it measured?

Mass is a measure of the amount of matter in an object. It is often measured in units such as grams or kilograms using a scale or balance.

What is density and how is it calculated?

Density is a measure of how tightly packed the particles in a substance are. It is calculated by dividing an object's mass by its volume, often expressed in units of grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3).

How are mass and density related?

Mass and density are directly proportional to each other. This means that as an object's mass increases, its density also increases, and vice versa. For example, a smaller object with the same mass as a larger object will have a higher density because its particles are more tightly packed.

How do you solve a basic mass and density problem?

To solve a basic mass and density problem, you will need to know the mass and volume of the object in question. Then, use the formula density = mass/volume to calculate the density. Make sure to use consistent units for mass and volume in your calculation.

What are some real-world applications of mass and density?

Mass and density are important concepts in various fields, including physics, chemistry, and engineering. They are used to understand the properties of materials, design structures and machines, and even study the composition of planets and stars. Some examples of real-world applications of mass and density include determining the purity of substances, predicting the behavior of fluids, and measuring the weight of objects.

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