How do I calculate center of mass using integrals and density?

In summary, the conversation discusses finding the x-bar of the function y=x3 using the equation \bar{x} = \frac{∫\tilde{x}dm}{∫dm}. The speaker asks if \tilde{x} represents the x distance times x3 and if x3 goes in the denominator before "dm". They also inquire about the same process for \tilde{y} and if a given density \rho stays in front of the integrands or in front of the entire equation. The question is then asked if the same process is done for \tilde{y} and if there are any differences.
  • #1
aaronfue
122
0
Just some things I need to verify:

When finding the x-bar of the function y=x3 using the equation:

[itex] \bar{x} [/itex] = [itex]\frac{∫\tilde{x}dm}{∫dm}[/itex],

Is my [itex]\tilde{x}[/itex] going to be the x distance(which will just be "x") times x3?

Also, will x3 be in the denominator just before "dm"?

The same process is done for the [itex]\tilde{y}[/itex], am I correct?

If I am also given a density [itex]\rho[/itex], will that stay in front of each integrand or in front of ([itex]\frac{∫\tilde{x}dm}{∫dm}[/itex])?

Is the same process is done for the [itex]\tilde{y}[/itex] or are there differences?

Thanks!
 
Engineering news on Phys.org

1. What is the Center of Mass?

The center of mass is a point in an object or system where the mass is evenly distributed in all directions. It is the average location of the mass of the object.

2. Why are Center of Mass calculations important?

Center of mass calculations are important because they help us understand the overall motion and stability of an object or system. They also play a crucial role in many physical principles, such as conservation of momentum and torque.

3. How do you calculate the Center of Mass?

The center of mass can be calculated by taking the weighted average of the positions of all the individual particles or objects in the system. This involves multiplying each particle's mass by its distance from a chosen reference point, and then dividing the sum of these products by the total mass of the system.

4. What is the significance of the Center of Mass being outside of an object?

If the center of mass of an object is outside of its physical boundaries, it means that the mass is unevenly distributed within the object. This can affect its stability and cause it to rotate or tip over.

5. Can the Center of Mass change?

Yes, the center of mass can change if the distribution of mass within the object or system changes. For example, if an object breaks into smaller pieces or if an external force acts on it, the center of mass will shift accordingly.

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