- #1
skrat
- 748
- 8
Homework Statement
Find the center of mass of Astroid ##x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} ## for ##x,y\geq 0## and ##a>0##.
Homework Equations
##x_T=\frac{\int x\left | \dot{\vec{r}}(t) \right |}{\int \left | \dot{\vec{r}}(t) \right |}##
The Attempt at a Solution
##x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}##
##(\frac{x}{a})^{\frac{2}{3}}+(\frac{y}{a})^{\frac{2}{3}}=1##
Now is it ok to say that ##\frac{x}{a}=\cos^3t## and ##\frac{y}{a}=\sin^3t## for ##t\in \left [ 0,\frac{\pi }{2} \right ]##?
Now ##\left | \dot{\vec{r}}(t) \right |=3a \sint \cost##.
Than ##x_T=\frac{\int x\left | \dot{\vec{r}}(t) \right |}{\int \left | \dot{\vec{r}}(t) \right |}## can be written as
##x_T=\frac{\int_{0}^{\frac{\pi }{2}} 3a^2\cos^4t\sint}{\int_{0}^{\frac{\pi }{2}} 3a\cost\sint}=\frac{2}{5}a##
or... is that parameterization wrong?