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Math Help - 3D Equilibrium of a rigid body

  1. #1
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    3D Equilibrium of a rigid body

    I am stuck on this question


    I generally know how to solve the problem but the force is throwing me off. Since it is parallel to the XY plane there will no be k component of the force. But how should I about breaking the 970N force into i,j components? Thanks.
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  2. #2
    Senior Member TriKri's Avatar
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    The distance from O to the point where \vec{F} takes effect is \vec{r} = (250,\ 200,\ -150)\ mm. \vec{F} is in the x-y plane, so \vec{F}=(F_x,\ F_y,\ 0). The momentum will then be

    \vec{M}=\vec{r}\times\vec{F}

    Now, we probably want to charge the support at O as much as possible, so we set |\vec{F}| = 970\ N. Besides, we want \vec{r} and \vec{F} to be orthogonal to each other, so

    \vec{r}\cdot\vec{F} = 0 \Leftrightarrow 250 F_x + 200 F_y = 0  \Leftrightarrow F_y = -\frac{5}{4}F_x

    F_z = 0, so \vec{F} is already in the x-y plane.

    So we could parametrisize \vec{F} and write \vec{F} = t\cdot(4,\ -5,\ 0), implying that |\vec{F}| = \sqrt{41}\ |t|, which means that \vec{F} could have basically any non-negative length.

    Now we know that there exist \vec{F} with the in the x-y plane with the length 970 N, orthogonal to \vec{r}. Let's use that in our momentum formula:

    \vec{M} = \vec{r}\times\vec{F} \Rightarrow

    |\vec{M}| = |\vec{r}\times\vec{F}| = |\vec{r}|\cdot|\vec{F}|\cdot\sin(\alpha) = \sqrt{250^2+200^2+(-150)^2}\ mm\ \cdot\ 970\ N\cdot\ \sin(\alpha) = \frac{485}{\sqrt{2}}\ Nm\cdot\sin(\alpha), where \alpha is the angle between \vec{r} and \vec{F}

    Now, since \vec{r} and \vec{F} are orthogonal, \sin(\alpha) = 1, so |\vec{M}| = \frac{485}{\sqrt{2}}\ Nm, which should be the maximum magnitude of the momentum.





    Alternativelly, you could turn \vec{F} into poolar coordinates:

    F_x=970\ N\cdot\cos(\varphi)
    F_y=970\ N\cdot\sin(\varphi)
    F_z=0

    perform the cross product between \vec{r} and \vec{F} = 970\ N\cdot(\cos(\varphi),\ \sin(\varphi),\ 0) to get the momentum, extract the magnitude of the momentum (the vector lenth), and analyse the expression to se which \varphi that gives the greatest magnitude of the momentum or to directly extract the greatest value of the expression.
    Last edited by TriKri; November 3rd 2008 at 11:14 AM.
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  3. #3
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    I got the answer, thanks man
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  4. #4
    Senior Member TriKri's Avatar
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    I'm studuing mechanics myself right now.
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