A uniform, thin wire is bent into the shape of a semicircle with a radius 'R'. Calculate the coordinates of the center of mass of this wire, assuming the mass per unit length is constant along the wire.
Question
A uniform, thin wire is bent into the shape of a semicircle with a radius 'R'. Calculate the coordinates of the center of mass of this wire, assuming the mass per unit length is constant along the wire.
Solution
The center of mass of a uniform, thin wire bent into the shape of a semicircle can be found by integrating over the mass distribution of the wire. However, because the wire is uniform, the center of mass will be located at the geometric center of the semicircle.
The geometric center of a semicircle is not the same as the center of the full circle from which it is derived. Instead, it is located along the line of symmetry of the semicircle, at a distance of 4R/(3π) from the flat base of the semicircle.
In a coordinate system where the flat base of the semicircle lies along the x-axis and the center of the full circle from which the semicircle is derived is at the origin, the coordinates of the center of mass are therefore (0, -4R/(3π)).
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