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How should I proceed to calculate the center of mass of objects like these two? In general I know how to calculate the center of mass but not for real objects like those.

enter image description here

enter image description here

The second one has a ball attached to it with a mass. I didn't give the numbers and all the givens because I am not just looking for the answers, I'm more looking for how I should proceed in general so that I can solve other similar problems

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  • $\begingroup$ Look for the centre of gravity. For irregular shapes, imagine hanging the object from a point on its edge and the centre of gravity will be found directly below the hanging point. Then do the same for more points, every time you do this, draw a vertical line downwards through each separate hanging point. Then think about what the point the lines cross at represents. $\endgroup$ – user167453 Oct 5 '17 at 13:51
  • $\begingroup$ This is interesting. I knew about that method and that the lines cross in the center of mass/gravity, but that's something we can do in real life. Suppose I am given numbers with these images. How should I proceed to find the center of mass/center of gravity algebraically? $\endgroup$ – Ayoub Rossi Oct 5 '17 at 14:41
  • $\begingroup$ hyperphysics.phy-astr.gsu.edu/hbase/cm.html#cmc. This uses the conventional approach of balancing torques, although without some symmetry based assumptions, I am guessing it will revolve (sorry:), around a system of simultaneous equations. My apologies, it's been a while since I covered this. $\endgroup$ – user167453 Oct 5 '17 at 15:46
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  1. Decide on a rectangular (Cartesian) co-ordinate system to use. For example, for the 3 legged stool the origin could be at the CM of the seat with one axis pointing downwards parallel to the legs.

  2. Divide the object into component parts which are regular 3D or 2D shapes for which it is easy to locate the CM. For example, cylinders, cuboids, spheres, cones, pyramids, circles, rectangles, triangles. Find the mass $m_i$ and co-ordinates of the CM $(x_i, y_i, z_i)$ for each component part in your co-ordinate system.

  3. The CM of the composite object $(\bar x, \bar y, \bar z)$ is the sum of the co-ordinates for each component shape, weighted by the fraction $\mu$ of the total mass in each component part :

$$\bar x= \mu_1 x_1+ \mu_2 x_2 + \mu_3 x_3 +...$$

and similar for $\bar y$ and $\bar z$, where $\mu_i=m_i/(m_1+m_2+m_3+...)$.

In step 2, if part of a regular shape is missing, treat the shape as two separate parts : (i) a complete shape with positive density and (ii) a missing shape with the same negative density. The missing shape will have a negative value for the weighting factor $\mu_i$.

See Calculating the Center of Mass.

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