# Center of mass of astroid

1. Jan 6, 2014

### skrat

1. The problem statement, all variables and given/known data
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 $a>0$.

2. Relevant equations

$x_T=\frac{\int x\left | \dot{\vec{r}}(t) \right |}{\int \left | \dot{\vec{r}}(t) \right |}$

3. 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?

2. Jan 6, 2014

### Dick

Looks fine to me.