Find Center of Mass for an open box Confused about steps for this problem

[Lo.Lee.Ta.]

Homework Statement

An appliance box has a square base with sides of length L, and has rectangular sides with a height of 3L. The top of the box is missing. The box is made from cardboard of uniform thickness and density. What is the height of the center of mass of this open box, with respect to the bottom of the box?

Homework Equations

Equation for the center of mass is: C.M.(x) = [(m1)(x1) + (m2)(x2) +...]/[m1 + m2 +..]
C.M.(y) = [(m1)(y1) + (m2)(y2) +...]/[m1 + m2 +..]

The Attempt at a Solution

This fellow on Yahoo Answers responded to this same question with the right answer. I am just confused about his solution... This is what he/she had written:

4 x 3L^2 x 3L/2 = 18L^3

total mass of box = 4 x 3L^2 + L^2 = 13L^2

18L^3/13L^2 = 18/13L

That is the right answer: 18/13L. But I just don't see how this method worked...
So the person squared the height of the box, and then multiplied that by 4. Why did he/she do that? That does not seem to be apart of the above formula... And then he multiplied that answer by L squared... Why are things even squared in order to solve the problem?

...As you can see, I don't get it at all... :/ Please explain this to me! Please help me to understand what each thing means in the solving of this problem!

Thank you so much for helping me! You are awesome! :)

 

[Dick]

1. An appliance box has a square base with sides of length L, and has rectangular sides with a height of 3L. The top of the box is missing. The box is made from cardboard of uniform thickness and density. What is the height of the center of mass of this open box, with respect to the bottom of the box?


2. Equation for the center of mass is: C.M.(x) = [(m1)(x1) + (m2)(x2) +...]/[m1 + m2 +..]
C.M.(y) = [(m1)(y1) + (m2)(y2) +...]/[m1 + m2 +..]

3. This fellow on Yahoo Answers responded to this same question with the right answer. I am just confused about his solution... This is what he/she had written:

4 x 3L^2 x 3L/2 = 18L^3

total mass of box = 4 x 3L^2 + L^2 = 13L^2

18L^3/13L^2 = 18/13L

That is the right answer: 18/13L. But I just don't see how this method worked...
So the person squared the height of the box, and then multiplied that by 4. Why did he/she do that? That does not seem to be apart of the above formula... And then he multiplied that answer by L squared... Why are things even squared in order to solve the problem?

...As you can see, I don't get it at all... :/ Please explain this to me! Please help me to understand what each thing means in the solving of this problem!

Thank you so much for helping me! You are awesome! :)

Write down what would be a value of m and x for each of the five parts of the box. You have one bottom and four identical sides. Then use your formula.

 

[PhanthomJay]

1. An appliance box has a square base with sides of length L, and has rectangular sides with a height of 3L. The top of the box is missing. The box is made from cardboard of uniform thickness and density. What is the height of the center of mass of this open box, with respect to the bottom of the box?


2. Equation for the center of mass is: C.M.(x) = [(m1)(x1) + (m2)(x2) +...]/[m1 + m2 +..]
C.M.(y) = [(m1)(y1) + (m2)(y2) +...]/[m1 + m2 +..]

3. This fellow on Yahoo Answers responded to this same question with the right answer. I am just confused about his solution... This is what he/she had written:

4 x 3L^2 x 3L/2 = 18L^3

total mass of box = 4 x 3L^2 + L^2 = 13L^2

18L^3/13L^2 = 18/13L

That is the right answer: 18/13L. But I just don't see how this method worked...
So the person squared the height of the box, and then multiplied that by 4. Why did he/she do that? That does not seem to be apart of the above formula... And then he multiplied that answer by L squared... Why are things even squared in order to solve the problem?

...As you can see, I don't get it at all... :/ Please explain this to me! Please help me to understand what each thing means in the solving of this problem!

Thank you so much for helping me! You are awesome! :)

You have the correct equation:

C.M.(y) = [(m1)(y1) + (m2)(y2) +...]/[m1 + m2 +..]
where m1, m2, etc. is the mass of one side of the box..there are 4 sides...and y is the distance fron the cm of each side to the reference axis at the bottom of the box. So assuming unit density, how would you calculate the mass of each side of the box?

 

[Lo.Lee.Ta.]

Hi! Thanks for responding! :)

Okay, so I was reading your explanation and trying to see what fit the formula:
m1(y1) + m2(y2).../(m1) + (m2) +...


So since there are 4 walls of the box that compose the height of the box with a mass of m, I wrote that the mass (m1) of the height is 4m.
Then I saw for the y1, the hight of the center of mass has to be half of the wall's height, so it is 3L/2.

So the equation so far is: (4m) + (3L/2)

And I compared that to the solution for the problem.
It looks like they are saying that m (the mass) equals the area for one wall of the box.
So then I found that m= (3L x L)= 3L^2

So for the denominator of the formula, it looks like m1 has to be the 3L^2 again (to represent the mass of the sides of the box) plus the mass of the base of the box (which is again the area). So that means the base of the box has a mass of L^2.

So putting the formula together, it is (3L^2 + (3L/2)) / (4(3L^2) + L^2) = 18L^3/13L^2=18/13L <-- Answer

*What confused me in this problem was the idea that the area of a side is really also the mass of the side! Still, how is this?

How is the area also the mass? I guess I get how the answer is gotten and how to do it again, but it still does not make any sense how the area is also the mass...

Could you or someone else please explain that to me? Thank you! :)

 

[haruspex]

So the equation so far is: (4m) + (3L/2)

Adding a mass to a length is not likely to produce anything meaningful.

 

[Lo.Lee.Ta.]

Oh, I typed it wrong... I meant times...
I'm using the formula: (m1)(y1) + (m2)(y2) +.../ (m1 + m2 +...)

I meant (4m)(3L/2) as the (m1)(y1).

And here I made another mistake:

"So putting the formula together, it is (3L^2 + (3L/2)) / (4(3L^2) + L^2) = 18L^3/13L^2=18/13L <-- Answer"

I left out the 4 and put in a + when it should have been a x.
I meant: (4 x 3L^2 x (3L/2)) / ((4 x 3L^2) + L^2) = 18/13L

This should be right now. But I'm still confused about how the area is also the mass...
Does anyone know why it is? Thanks!

 

[haruspex]

total mass of box = 4 x 3L^2 + L^2
So the person squared the height of the box, and then multiplied that by 4.

No. It's 3L^2 (the area of one side of the box), not (3L)^2. And 4 sides.

 

[PhanthomJay]

But I'm still confused about how the area is also the mass...
Does anyone know why it is? Thanks!

Mass is not area, mass is volume times density. However, from the problem statement,

The box is made from cardboard of uniform thickness and density

. That should help.