Parametric Centroid Problem

May 2010
2
0
So one of the things I have to know for my Calc final is how do find the center of mass of a centroid given its parametric formulas. We learned how to find centers of mass and how to use parametric equations but never put the two together in class.
So how would I go about solving a problem like this:

Find the y-coordinate of the centroid oft he curve given by the parametric equations:
x=(3^(1/2))t^2
y=t-t^3

t= [0, 1]

I can figure out most of it, I'm just stuck trying to figure out how I would find the area of the centroid

Thanks :)
 
Apr 2010
384
153
Canada
So one of the things I have to know for my Calc final is how do find the center of mass of a centroid given its parametric formulas. We learned how to find centers of mass and how to use parametric equations but never put the two together in class.
So how would I go about solving a problem like this:

Find the y-coordinate of the centroid oft he curve given by the parametric equations:
x=(3^(1/2))t^2
y=t-t^3

t= [0, 1]

I can figure out most of it, I'm just stuck trying to figure out how I would find the area of the centroid

Thanks :)
The basic form of C.O.M is

\(\displaystyle \bar x = \frac{ \iiint x * (density) dx }{ \iiint (density) dx } \)

The same goes for y and z bar. Nothing changes in parametric form, we simply compute the integral. In this case,

\(\displaystyle \bar x = \frac { \int_0^1 3^{1/2} t^2 * density dt } {mass} \)

\(\displaystyle \bar y = \frac { \int_0^1 (t-t^3)*density dt } {mass} \)

I would like to point out that you cannot find the C.O.M without knowing the mass or the density of the shape. But the above is how you calculate the required components. The numerators are of course Mx=0 and My=0.
 
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