1. ## algebraic manipulation

Hello,

I'm trying to follow the rearranging of a formula from:

$\displaystyle M=\frac {m\Omega^2}{\sqrt{r^2 \Omega^2+(k-m \Omega^2)^2}}$

to:

$\displaystyle M=\frac {\beta^2}{\sqrt{(1-\beta^2)^2+4\alpha^2\beta^2}}$

Using the following;

$\displaystyle \alpha=\frac{r}{2\sqrt{mk}}$

$\displaystyle \omega = \sqrt{k/m}$

$\displaystyle \beta=\Omega/\omega$

I have got as far as;

$\displaystyle M=\frac {m \Omega^2}{\sqrt{4 \alpha^2km \Omega^2+k^2-m^2 \Omega^4}}$

but always have unwanted variables leftover... any help please.

2. Originally Posted by robocombot
Hello,

I'm trying to follow the rearranging of a formula from:

$\displaystyle M=\frac {m\Omega^2}{\sqrt{r^2 \Omega^2+(k-m \Omega^2)^2}}$

to:

$\displaystyle M=\frac {\beta^2}{\sqrt{(1-\beta^2)^2+4\alpha^2\beta^2}}$

Using the following;

$\displaystyle \alpha=\frac{r}{2\sqrt{mk}}$

$\displaystyle \omega = \sqrt{k/m}$

$\displaystyle \beta=\Omega/\omega$

I have got as far as;

$\displaystyle M=\frac {m \Omega^2}{\sqrt{4 \alpha^2km \Omega^2+k^2-m^2 \Omega^4}}$

but always have unwanted variables leftover... any help please.
Hi

You can see that $\displaystyle \Omega$ is present only in $\displaystyle \beta$ expression therefore you can replace $\displaystyle \Omega$ by $\displaystyle \beta \omega$

and that $\displaystyle r$ is present only in $\displaystyle \alpha$ expression therefore you can replace $\displaystyle r$ by $\displaystyle 2\alpha \sqrt{mk}$

$\displaystyle M=\frac {m\Omega^2}{\sqrt{r^2 \Omega^2+(k-m \Omega^2)^2}}$

$\displaystyle M=\frac {m\beta^2 \omega^2}{\sqrt{4 \alpha^2 m k \beta^2 \omega^2+(k-m \beta^2 \omega^2)^2}}$

Dividing numerator and debominator by $\displaystyle m \omega^2$

$\displaystyle M=\frac {\beta^2}{\sqrt{4 \alpha^2 \beta^2 \frac{k}{m \omega^2}+(\frac{k}{m \omega^2}- \beta^2)^2 }}$

Using $\displaystyle \frac{k}{m \omega^2} = 1$

$\displaystyle M=\frac {\beta^2}{\sqrt{(1-\beta^2)^2+4\alpha^2\beta^2}}$

3. thanks, great help!

4. Hello, robocombot!

Show that:
. . $\displaystyle M \;=\;\dfrac {m\Omega^2}{\sqrt{r^2 \Omega^2+(k-m \Omega^2)^2}} \;=\;\dfrac {\beta^2}{\sqrt{(1-\beta^2)^2+4\alpha^2\beta^2}}$

using the following; .$\displaystyle \begin{Bmatrix} \alpha &=&\frac{r}{2\sqrt{mk}} & [1] \\ \\[-3mm] \omega &=& \sqrt{\frac{k}{m} & [2] \\ \\[-3mm] \beta&=&\frac{\Omega}{\omega}& [3] \end{Bmatrix}$

$\displaystyle \text{From [2] and [3]: }\;\beta \:=\:\dfrac{\Omega}{\sqrt{\frac{k}{m}}} \:=\:\Omega\sqrt{\dfrac{m}{k}} \quad\Rightarrow\quad \beta^2 \:=\:\dfrac{m\Omega^2}{k}$

I started with the right side . . .

$\displaystyle M \;=\;\dfrac{\beta^2}{\sqrt{(1-\beta^2)^2 + 4\alpha^2\beta^2}}$

. . .$\displaystyle \;=\; \dfrac{ \dfrac{m\Omega^2}{k}} {\sqrt{\left(1-\dfrac{m\Omega^2}{k}\right)^2 + 4\left(\dfrac{r^2}{4mk}\right)\left(\dfrac{m\Omega ^2}{k}\right)}}$

. . .$\displaystyle =\; \frac{ \dfrac{m\Omega^2}{k}} {\sqrt{\left(\dfrac{k-m\Omega^2}{k}\right)^2 + \dfrac{r^2\Omega^2}{k^2}}}$

. . .$\displaystyle =\; \frac{\dfrac{m\Omega^2}{k}} {\dfrac{\sqrt{(k-m\Omega^2)^2 + r^2\Omega^2}}{k}}$

. . .$\displaystyle =\;\dfrac{m\Omega^2}{\sqrt{r^2\Omega^2 + (k-m\Omega^2)^2}}$