# How Do You Calculate the Center of Mass with Non-Constant Density Using Vectors?

• Ataman
In summary: That doesn't save any work.In summary, the conversation discusses finding the center of mass of an object defined by the functions x^{2} and \sqrt{x}, without computing x and y separately. The conversation also mentions the use of density as a varying function and the method of using a scalar product instead of a dot product when dealing with a varying density. However, it is noted that this method does not necessarily save any work.
Ataman
I am looking for a way to find the center mass of an object whose area is enclosed by $$x^{2}$$ and $$\sqrt{x}$$ without computing the x and y seperately (a great deal of paperwork).

So...

$$M\overrightarrow{R_{cm}} = \int \overrightarrow{r} dm$$

$$\sigma = \frac{M}{A} = \frac{dm}{dA}$$

$$\sigma A \overrightarrow{R_{cm}} = \int \sigma \overrightarrow{r} dA$$

$$\overrightarrow{R_{cm}} = \frac{\int\int \sigma \overrightarrow{r} dy dx } {\int \sigma (f(x)-g(x))dx}$$

Because they are constants, the sigmas cancel and I eventually end up with...

$$\overrightarrow{R_{cm}} = \frac{\int^1_0\int^{x^{2}}_{\sqrt{x}} (xi+yj) dydx}{\int^1_0 x^{2} - \sqrt{x} dx}$$

(The answer is $$\frac{9}{20}i + \frac{9}{20}j$$)

But what happens when sigma/density is not constant, but is given a value say... xi or something like that? Obviously taking the dot product will not work, and I am unsure about the cross product (I haven't done a lot of vectors).

-Ataman

What do you mean that the dot product won't work? I don't see any dot product in the above derivation.

In the above derivation, the density is constant, so it is not defined by a vector.

What I am looking for is a case where there is a varying density within the region.

-Ataman

Ataman said:
In the above derivation, the density is constant, so it is not defined by a vector.

What I am looking for is a case where there is a varying density within the region.

-Ataman

Why define it as a vector? Why not make it as a function of the position vector?

That's what I meant. Excuse me.

-Ataman

Ataman said:
That's what I meant. Excuse me.

-Ataman

Then don't you know the answer?

Then it is still not a vector. "Density" is a numeric value, not a vector function. You don't need a "dot product", you just have a scalar product- multiply each component of the position vector by the density function.

You should notice that you aren't really doing less work that if you did x and y as separate integrals. Since $\int u\vec{i}+ v\vec{j} dx= \int u\vec{i}dx+ \int v\vec{j}dx$ you are just writing two integrals as if they were one.

## What is the formula for finding the center of mass using vectors?

The formula for finding the center of mass using vectors is:

CM = (m1*r1 + m2*r2 + ... + mn*rn) / (m1 + m2 + ... + mn)

Where CM is the center of mass, m is the mass of each object, and r is the position vector of each object.

## How is center of mass different from center of gravity?

Center of mass refers to the point where the mass of an object or system is equally distributed, while the center of gravity refers to the point where the weight of an object or system is equally distributed. In most cases, the center of mass and center of gravity are located at the same point, but they can differ in situations where gravity is not uniform (e.g. near a black hole).

## Can the center of mass be outside of an object?

Yes, the center of mass can be outside of an object. This can occur when the object has an irregular shape or when there are external forces acting on the object. In these cases, the center of mass will be located at the point where the object's mass is evenly distributed.

## How is center of mass used in physics?

The concept of center of mass is used in many areas of physics, including mechanics, astronomy, and thermodynamics. It is used to determine the overall behavior of a system, such as the motion of objects under the influence of external forces, the stability of structures, and the distribution of energy in a system.

## What is the significance of the center of mass in sports?

In sports, the center of mass is important in maintaining balance and stability. Athletes must be aware of their center of mass and how it shifts during different movements in order to maintain control and prevent falling. For example, in figure skating, the center of mass must be kept over the skates to maintain balance and execute complex moves.

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