Does Changing Material Density Affect the Center of Mass in a Composite Object?

[1MileCrash]

My textbook is giving awfully complicated formulas for centers of mass for actual objects (if they are a system of massive points, that's simpler for me.)

Intuitively though, it just seems like it would be a weighted average??

So,

If I have a solid metal cube, divided into two halfs, one iron, and one with a density of less than that of iron, does it have the same center of mass as if the cube was pure iron, but the half that was "formerly" a less dense metal was "smushed" along the axis perpendicular to the line the cube was divided among so that it has the same mass as it did when the lighter metal was there?

 

[daqddyo1]

The centre of mass of an object is the point at which it can be balanced (in a gravitational field). It is usually difficult though to balance something exactly at its C of M as it is usually inside the object.
If you could balance the object on a point, then the C of M would be directly above that point.
If you suspend the object using a (massless) string, then the C of M would be directly below and in line with the string.
In your examples, think about trying to balance the cube at different points on its surface. What would you discover?

Does the C of M stay fixed or does its position change?

A neat C of M experiment is to take two forks and stick them together at their tines so that the reulting shape is a rough parabola. Where is the centre of mass of the pair of forks?
To find it, determine where it can be balanced - by sticking a toothpick into the forks so that you can balance them by resting the other end of the toothpick on your finger.

 

[sophiecentaur]

My textbook is giving awfully complicated formulas for centers of mass for actual objects (if they are a system of massive points, that's simpler for me.)

Intuitively though, it just seems like it would be a weighted average??

So,

If I have a solid metal cube, divided into two halfs, one iron, and one with a density of less than that of iron, does it have the same center of mass as if the cube was pure iron, but the half that was "formerly" a less dense metal was "smushed" along the axis perpendicular to the line the cube was divided among so that it has the same mass as it did when the lighter metal was there?

The CM of the two different blocks can be calculated by first working out where the CMs of the two individual blocks is. Then using those two CMs and the two masses, you can work out the resulting CM position. That's not too complicated.
Many of these problems can be made simpler by doing them in stages.

Your "weighted average" description is just about exactly what I described.

 

[256bits]

1Milecrash,
Just take note that the CM is computed for the x, y and z axis seperately, and where they meet is the real CM. Take a donut - CM right in the middle of the donut since it is symetrical on all 3 axis. Half a donut - well it is symmetrical on 2 axis but not the third, so you have to do your calculation for that 3rd axis. Most statistics books at the back have tables listing different shapes and you can use these to obtain your weighted averages. If it is a really oddball shape than you just have to go through with all the calculations

 

[pmacias]

The center of mass is a concept used to describe the balance point of an object or system. It is the point at which the object can be considered to have all of its mass concentrated. In simpler terms, it is the point at which the object would balance if it were placed on a pivot or fulcrum.

You are correct in thinking that the center of mass can be thought of as a weighted average. It is calculated by taking into account the mass and position of each individual particle or component of an object.

In your example of the metal cube, the center of mass would not change if the lighter metal was "smushed" to have the same mass as the iron. This is because the overall mass and the distribution of that mass within the object have not changed. However, if the lighter metal was removed and replaced with a heavier material, the center of mass would shift towards the heavier side.

It is important to note that the center of mass may not always coincide with the geometric center of an object. This is because the distribution of mass within an object can vary and affect the location of the center of mass.

I hope this helps clarify the concept of center of mass for you. Remember, it is just a way to describe the balance point of an object and can be calculated using the mass and position of each individual component.