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System of Equations with Two Unknowns, Algebra Help

  1. Dec 31, 2013 #1
    1. The problem statement, all variables and given/known data

    I have two equations with two unknowns. I know m, F, R, r and I. I need to find a and f.

    m = 2
    F = 28
    R = 0.25
    I = 0.0625
    r = 0.1875

    I know the ultimate answers are [itex]a = 16\frac{1}{3}[/itex] and [itex]f = 4\frac{2}{3}[/itex]

    2. Relevant equations

    (1) [itex]Fr-fR=I\frac{a}{R}[/itex]

    (2) F+f=ma

    3. The attempt at a solution

    [itex]Fr-I\frac{a}{R}=fR[/itex]

    [itex]f=\frac{Fr}{R}-I\frac{a}{R^2}[/itex]

    [itex]I(1+\frac{r}{R})=a(m+\frac{I}{R^2})[/itex]

    [itex]a=F(\frac{1+\frac{r}{R}}{m+\frac{I}{R^2}})[/itex]

    [itex]28(\frac{1+\frac{0.1875}{0.25}}{2+\frac{0.0625}{0.25^2}})[/itex] = ?

    16.333 (Yay!)

    I can't get back to f from here. :( Help!
     
    Last edited: Dec 31, 2013
  2. jcsd
  3. Dec 31, 2013 #2

    LCKurtz

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    From (2) ##f = ma - F## so plug your numbers in.
     
  4. Jan 1, 2014 #3
    Ok, fair enough. But the answer guide said you could arrive at the following equation for f and I want to know how they got it:

    [itex]f=\frac{\frac{r}{R}-\frac{I}{mR^2}}{1+\frac{I}{mR^2}}[/itex]
     
  5. Jan 1, 2014 #4

    ehild

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    F is missing from the expression for f. It should be

    [itex]f=F\frac{\frac{r}{R}-\frac{I}{mR^2}}{1+\frac{I}{mR^2}}[/itex]

    Plug in your equation [itex]a=F(\frac{1+\frac{r}{R}}{m+\frac{I}{R^2}})[/itex] for a into the equation f+F=ma.

    ehild
     
  6. Jan 1, 2014 #5

    vela

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    In matrix form, your system of equations is
    $$\begin{pmatrix} R & \frac{I}{R} \\ -1 & m \end{pmatrix}\begin{pmatrix} f \\ a \end{pmatrix} = \begin{pmatrix} Fr \\ F \end{pmatrix}$$ A good technique in cases like this one is to use Cramer's rule. It'll get you the solution with minimal algebra.

    http://en.wikipedia.org/wiki/Cramer's_rule#Applications
     
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