This discussion focuses on calculating the centroid and moments of inertia for a circle with a surface density that varies as the nth power of the distance from a point O on the circumference. The centroid's x-coordinate is derived as x' = 2a(n-2)/(n+2), while the moment of inertia about the diameter through O involves integrating the expression -4(a×a)M. The calculations utilize polar coordinates, specifically the equation r = 2cos(θ), and the bounds for integration are set between -π/2 and π/2 for θ, and 0 to 2cos(θ) for r.
PREREQUISITESMathematicians, physicists, and engineering students who are involved in mechanics and need to understand the properties of shapes with varying density distributions.
Thanks a lot!Infrared said:Working in coordinates, we can assume that ##O## is the origin, and the circle is ##(x-1)^2+y^2=1##. In polar coordinates, this circle is ##r=2\cos(\theta)##. The ##y##-coordinate of the center of mass is ##0## by symmetry, and the ##x##-coordinate is the integral of ##x r^n dx dy=\cos\theta \ r^{n+2} dr d\theta ## over the disk, divided by the mass, which should be straightforward to calculate (bounds are ##-\pi/2\leq\theta\leq\pi/2## and ##0\leq r\leq 2\cos\theta##).
The moment of inertia calculations are similar.