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Math Help - algebraic manipulation

  1. #1
    Newbie
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    Jun 2010
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    algebraic manipulation

    Hello,

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

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

    to:

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

    Using the following;

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

    \omega = \sqrt{k/m}

    \beta=\Omega/\omega

    I have got as far as;

    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.
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
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    France
    Posts
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    Quote Originally Posted by robocombot View Post
    Hello,

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

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

    to:

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

    Using the following;

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

    \omega = \sqrt{k/m}

    \beta=\Omega/\omega

    I have got as far as;

    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 \Omega is present only in \beta expression therefore you can replace \Omega by \beta \omega

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

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

    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 m \omega^2

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

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

    M=\frac {\beta^2}{\sqrt{(1-\beta^2)^2+4\alpha^2\beta^2}}
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  3. #3
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    thanks, great help!
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  4. #4
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    Lexington, MA (USA)
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    Hello, robocombot!

    Show that:
    . . 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; . \begin{Bmatrix}<br />
\alpha &=&\frac{r}{2\sqrt{mk}} & [1]  \\ \\[-3mm]<br />
\omega &=& \sqrt{\frac{k}{m} & [2] \\ \\[-3mm]<br />
\beta&=&\frac{\Omega}{\omega}& [3] \end{Bmatrix}

    \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 . . .


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


    . . . \;=\; \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)}} <br />


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


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


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

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