# How to Locate the Center of Gravity

1. Apr 22, 2012

### siukwok

Hi,

I have some questions regarding on the locating of the center of gravity. I have read some txtbk regarding on finding the COG for structure/element that is homogeneous. Therefore, the COG will conincide with the centroid of the the volume. Am i right to say that homogeneous structure/element means that it is made up of the same material? How am i supposed to find the COG if the element is made up of different materials? Lastly, if i know the COG of several components (of different materials) which are subsequently fixed together, am i right to say that it has to be treated as a problem in relation to non-homogeneous element? My boss has tasked me to find out how this is done but i have difficulties finding relevant reading materials for the above problems. Your help is greatly appreciated.

Thanks!!

Siukwok

2. Apr 22, 2012

### HallsofIvy

Staff Emeritus
Yes, saying that an object is "homogeneous" means it made of the same material throughout so, in particular, the density is constant. if the objecft is not homogeneously but the density varies continuously, and the geometry of the figure is reasonably simple, you can try integrating the density to find the mass: $M= \int\int\int_V \rho dV$ where $\rho$ is the density funcdtion.

Then the x, y, and z coordinates of the center of mass are given by
$$\overline{x}= \int\int\int x\rho dV/M$$
$$\overline{y}= \int\int\int y\rho dV/M$$
$$\overline{z}= \int\int\int z\rho dV/M$$

If the the object can be divided into several pieces, each being uniform, you can find the center of mass of each piece then form a "weighted" average of the coordinates, weighted by the mass of each piece.

3. Apr 22, 2012

### siukwok

Hi HallsofIvy,

Thanks for ur swift reply. However, i probably need some examples to get a better idea of how it is done. I have sketched an object made up of 2 cuboids as attached. The one on top is of 50x70x55 mm^3 and the one below is 100x150x40 mm^3. Say the density for the cuboid on top is 7800 kg/m^3 and the density for the cuboid at the bottom is 7000 kg/m^3. The COG for both cuboids are apparently at the center of the respective cuboids. How am i supposed to locate the COG of the object as a whole?

Thanks!

Siukwok

#### Attached Files:

• ###### Part1.JPG
File size:
25.5 KB
Views:
82
4. Apr 22, 2012

wish you were able to get it worked out!

5. Apr 23, 2012

### chrisbaird

First find the center of mass of each block. For a symmetric object made of the same material, its just the geometric center. Now fix the blocks together and write their center locations in terms of some overall coordinate system. (for instance, if two blocks are combined and then place flat on a table, you would say the upper block's center is 3 inches from the tabletop, y2 = 3, and the bottom block's center is 0.5 inch from the tabletop, y1 = 0.5; the upper block's center is 5 inches from the table edge, x2 = 5, etc.) Also measure the mass of each block. The center of mass of the two blocks combined is then:

x = (m1 x1 + m2 x2)/(m1 + m2)

y = (m1 y1 + m2 y2)/(m1 + m2)

z = (m1 z1 + m2 z2)/(m1 + m2)

This is what HallsofIvy meant by a weighted average, but this is really just a simplified version of the integrals he gave.

6. Apr 24, 2012

### siukwok

Thank you so much! =)