What Is the Center of Mass Position for Three Aligned Cubes?

[Bones]

Homework Statement

Three cubes, of side L1, L2, and L3, are placed next to one another (in contact) with their centers along a straight line as shown in the figure. What is the position, along this line, of the CM of this system? Assume the cubes are made of the same uniform material and L1= 3.5 cm.

http://www.webassign.net/gianpse4/9-44.gif

Homework Equations

The Attempt at a Solution

Xcm=73.5cm^3(d)*1.75cm+294cm^3(d)*7cm+661.5cm^3(d)+15.75cm/73.5(d)+294(d)+661.5(d)
Xcm=12.25
This is not correct and I am not sure what I am doing wrong. I got the area of each cube by multiplying 6*side^2 and then multiplied that by each location of the center of mass and divided by sum of the masses. Please help!

 

[Bones]

Never mind, I figured it out. I was using area and I needed to use volume for a 3D shape.

 

[thomate1]

Hello,

I would first like to clarify that the center of mass is a point in a system where the mass is evenly distributed in all directions. It is the point where the system can be balanced on a pivot without any rotation occurring.

To solve this problem, we can use the formula for the center of mass of a system of particles:

Xcm = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)

Where Xcm is the position of the center of mass along the line, m1, m2, and m3 are the masses of each cube, and x1, x2, and x3 are the positions of their respective centers of mass along the line.

Since all the cubes are made of the same material and have the same dimensions, we can assume that they have the same mass. Therefore, we can simplify the formula to:

Xcm = (x1 + x2 + x3) / 3

Now, we need to find the positions of the centers of mass, x1, x2, and x3. We can do this by dividing the length of each cube by 2, as the center of mass of a cube is located at its geometric center. So, we get:

x1 = L1/2 = 3.5/2 = 1.75 cm
x2 = L2/2 = 3.5/2 = 1.75 cm
x3 = L3/2 = 3.5/2 = 1.75 cm

Substituting these values into the simplified formula, we get:

Xcm = (1.75 + 1.75 + 1.75) / 3 = 1.75 cm

Therefore, the position of the center of mass of this system is located at 1.75 cm along the line. This makes sense intuitively, as all the cubes are of equal mass and evenly distributed along the line, so the center of mass is located in the middle of the line.

I hope this helps clarify the concept of center of mass and how to solve this problem. Let me know if you have any further questions. Keep up the good work in your studies!