Solving Center of Mass Doubt: What is 'y' in Integration?

In summary, the conversation is discussing the method of integration to find the center of mass for bodies with continuous distribution of mass. The coordinates x and y represent the location of the mass element dm and the center of mass can be thought of as the average location of all the mass. The position of the center of mass for a semi-circular ring can be found by looking it up or by practicing the integration method.
  • #1
rahul.mishra
7
0
A silly doubt regarding center of mass...
As we know for bodies having continuous distribution of mass we can know their center of mass by the method of integration...
like, Xcm = 1/M∫x.dm

but what is x here?
in many cases...
like in finding the COM of a ring
Xcm = 0 and Ycm = 2r/∏ (ofcouse when the center is taken on the mid point of diameter)

but on finding the COM of a semi-circular disc or plate...
we can assume it to be formed with different rings...
finally in the formula Ycm = 1/M∫y.dm we set y = 2r/pi ...
so i just can't understand what this 'y' is... and how do we decide it in different cases...
i think you can get what i want to ask actually...
and please help me... i m new to this site...
rahul :)
 
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  • #2
I'm not sure I get your question, but x and y represent the coordinates of the mass element dm. And the center of mass, (Xcm, Ycm), can be thought of as the 'average' location of all the mass.
 
  • #3
can you tell me the position of the center of mass of a ring of radius R and mass M?
Does it lie on the circumference or somewhere else when origin is assumed to be at the
center of the diameter joining the two ends of the ring?
 
  • #4
rahul.mishra said:
can you tell me the position of the center of mass of a ring of radius R and mass M?
Does it lie on the circumference or somewhere else when origin is assumed to be at the
center of the diameter joining the two ends of the ring?

The center of mass (assuming uniform density) of a circle (ring, torus) is at the center of the circle.
 
  • #5
Oh...! i m really sorry... i meant a semi-circular ring...!
 
  • #6
rahul.mishra said:
Oh...! i m really sorry... i meant a semi-circular ring...!
You can just look it up. (Like here: List of centroids.) But you'd better practice so you can do the integration on your own.
 

Related to Solving Center of Mass Doubt: What is 'y' in Integration?

1. What is the concept of center of mass in physics?

The center of mass is a point in an object or system that represents the average position of all its mass. It is the point at which an object can be balanced, and it follows the laws of motion and conservation of momentum.

2. How is the center of mass calculated?

The center of mass can be calculated by finding the weighted average of the positions of all the individual particles or objects in a system. This is typically done using the formula xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn), where x represents position and m represents mass.

3. What is the role of 'y' in integration when solving for center of mass?

The 'y' in integration represents the vertical position of an object. In finding the center of mass, both the horizontal and vertical positions must be taken into account, so 'y' is used in the integration process to calculate the weighted average of the vertical positions.

4. Can the center of mass be outside the physical boundaries of an object?

Yes, the center of mass can be outside the physical boundaries of an object. This typically occurs when the object has an irregular shape or when the distribution of mass is uneven. In these cases, the center of mass may be located at a point outside the object's physical boundaries.

5. How does the center of mass affect an object's stability?

The center of mass plays a crucial role in an object's stability. If the center of mass is located within the object's base of support, the object will be stable. However, if the center of mass is located outside the base of support, the object will be unstable and may tip over. This is why it is important to calculate and understand the center of mass in various systems and objects.

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