- #1
mnb96
- 715
- 5
Hello,
I was thinking of the following question: suppose we have a body in two-dimensional space and its density is described by the function [itex]f(x,y)[/itex] (Note! in Cartesian coordinates!). The x-coordinate of the center of mass is given by [tex]C_x= \frac{\int\int xf(x,y)dxdy}{\int\int f(x,y)dxdy}[/tex] where the domain of integration is ℝ2. The y-coordinate of the center of mass is given by an analogous formula.
Now, suppose we are given the density of this body in polar coordinates, that is: f(r,θ). Is it possible to obtain directly the centroid [itex](C_r, C_\theta)[/itex] in polar coordinates from f(r,θ)?
Of course I know that by using a polar-to-Cartesian curvilinear transformation x=x(r,θ), y=y(r,θ), we can easily obtain the formulas [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex], and then we could just convert Cx and Cy into polar coordinates. However if you think about it, when we calculate [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex] we are essentially calculating the Cartesian coordinates of the center of mass.
Can we "bypass" this step? What's so special here with Cartesian coordinates? Aren't we supposed to be able to calculate the center of mass of a body without resorting to a special coordinate system?
I was thinking of the following question: suppose we have a body in two-dimensional space and its density is described by the function [itex]f(x,y)[/itex] (Note! in Cartesian coordinates!). The x-coordinate of the center of mass is given by [tex]C_x= \frac{\int\int xf(x,y)dxdy}{\int\int f(x,y)dxdy}[/tex] where the domain of integration is ℝ2. The y-coordinate of the center of mass is given by an analogous formula.
Now, suppose we are given the density of this body in polar coordinates, that is: f(r,θ). Is it possible to obtain directly the centroid [itex](C_r, C_\theta)[/itex] in polar coordinates from f(r,θ)?
Of course I know that by using a polar-to-Cartesian curvilinear transformation x=x(r,θ), y=y(r,θ), we can easily obtain the formulas [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex], and then we could just convert Cx and Cy into polar coordinates. However if you think about it, when we calculate [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex] we are essentially calculating the Cartesian coordinates of the center of mass.
Can we "bypass" this step? What's so special here with Cartesian coordinates? Aren't we supposed to be able to calculate the center of mass of a body without resorting to a special coordinate system?