Finding the center of mass of an incomplete circle

In summary, the task is to locate the x coordinate of the center of mass of a homogeneous rod bent into a circular arc with a radius of 170mm. Using the equations x=rcosθ, y=rsinθ, and dL=r*dθ, the center of mass is found by integrating "170 dθ" and "170 (cosθ) 170 dθ" from -2π/3 to +2π/3, giving a result of 70.3mm.
  • #1
Jbray
3
0

Homework Statement



Locate the x coordinate of the center of mass of the homogeneous rod bent into the shape of a circular arc. Take r = 170 .

The arc goes from (-5/6) to (5/6)pi (counterclockwise). It has a radius of 170mm.

Homework Equations



x=rcosθ, y=rsinθ, dL=r*dθ

The Attempt at a Solution



I found "M" by integrating "170 dθ" from (-5/6)pi to (5/6)pi. This gave me 890.12mm.I found "My" by integrating "170 (cosθ) 170 dθ" from (-5/6)pi to (5/6)pi. This gave me 170^2 or 28900mm.

I used My/M to find the x coordinate of the center of mass as 28900/890.12 or 32.468mm. However this is incorrect.
 
Last edited:
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  • #2
The solution looks good, but you miss the unit, and try to give the result with three significant digits.

ehild
 
  • #3
I updated the units. I also found out that the answer is supposed to be 70.3mm but I still can't figure out how to get that.
 
  • #4
Are the limits -5/6 and 5/6pi, or -5/6pi and 5/6pi? But even then, the result is different from that 70 mm. The given results happen to be wrong quite often.

ehild
 
  • #5
Jbray said:
the answer is supposed to be 70.3mm
That would be the answer if the angle were -2π/3 to +2π/3.
 
  • #6
haruspex said:
That would be the answer if the angle were -2π/3 to +2π/3.

You are a genius! So they took over the solution from an old version of the problem, while changing the limits in the problem text. :biggrin:

ehild
 

1. How do you find the center of mass of an incomplete circle?

The center of mass of an incomplete circle can be found by using the equation: xcm = (A1x1 + A2x2 + A3x3 + ...)/A, where x1, x2, x3, etc. are the x-coordinates of the individual segments of the circle, A1, A2, A3, etc. are their respective areas, and A is the total area of the incomplete circle.

2. Can the center of mass of an incomplete circle be outside of the circle?

Yes, the center of mass can be located outside of the circle if the segments of the circle are not evenly distributed. This means that the center of mass may not always coincide with the geometric center of the circle.

3. What is the significance of finding the center of mass of an incomplete circle?

The center of mass is an important concept in physics, as it represents the point where an object's mass can be considered to be concentrated. In the case of an incomplete circle, finding the center of mass can help determine the balance and stability of the object.

4. What factors can affect the location of the center of mass in an incomplete circle?

The location of the center of mass in an incomplete circle can be affected by the distribution of mass along the circumference of the circle, as well as the shape and size of the individual segments of the circle. Any changes in these factors can result in a shift in the center of mass.

5. Is there a difference between the center of mass and the center of gravity in an incomplete circle?

Yes, there is a difference between the center of mass and the center of gravity in an incomplete circle. The center of mass is a geometric property that takes into account the distribution of mass, while the center of gravity takes into account the effects of gravity on the object. In most cases, the center of mass and the center of gravity will be at the same location, but in some cases, they may differ.

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