Calculating Gravitational Force on a Point Mass in a Semicircular Rod

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To calculate the gravitational force exerted by a semicircular rod on a point mass at its center of curvature, it's essential to consider the contributions from each infinitesimal mass segment along the rod. The symmetry of the semicircle means that the horizontal components of the gravitational forces will cancel out, leaving only the vertical components to be summed. This can be achieved by breaking the rod into small mass elements (dm) and integrating the y components of the forces they exert. The integration is most effectively performed by converting the mass distribution into an angle-based format. Ultimately, this approach will yield the total gravitational force acting on the point mass.
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A uniform rod of mass M = 20kg and length L = 5m is bent into a semicircle. What is the gravitational force exerted by the rod on a point mass m = 0.1 kg located at the center of curvature of the circular arc?
 
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I have assumed that this must be solved using the summation of all points on the rod since each exerts its own force on the point mass. Assuming that to be true, I figured that the x components all cancel each other out, leaving only the y components. But now, I don't know how to solve this?
 
You are on the right track. You have the symmetry figured out. Break the rod up into bits of mass dm and write the y component of the force related to each bit of mass. Add all the y components (integrate over the masses). This is most easily done by converting to an integral over an angle.
 
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