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Thread: mechanics, vectors components help

  1. #1
    Super Member
    Sep 2008

    mechanics, vectors components help

    express the force vectors in terms of components parallel and perpendicular to the plane shown.

    Can I say the angle the particle make with the plane is also 43 degrees , because they are alternate angles?

    how do I make a triangle with the 10N force being the hypotenuse from this diagram ?

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  2. #2
    Super Member Aryth's Avatar
    Feb 2007
    You're going to want to keep the axes as is, but a quick translation shows that you are correct in assuming the $\displaystyle 43^{\circ}$ angle.

    Now, you may or may not know that the components of the vectors are scalars, represented by $\displaystyle F_x$ for the force in the $\displaystyle \hat{i}$ direction and $\displaystyle F_y$ for the force in the $\displaystyle \hat{j}$.

    We also know that the magnitude of the force (I'll call $\displaystyle F_T$), which is 10N, is the hypotenuse of a right triangle, meaning that we can define $\displaystyle \cos{\theta}$ and $\displaystyle \sin{\theta}$ in terms of the force:

    $\displaystyle \cos{\theta} = \frac{F_x}{F_T}$

    Because in the unit circle, we define it to be cos = x/hyp.

    This means that:

    $\displaystyle F_x = F_T\cos{\theta}$


    $\displaystyle F_y = F_T\sin{\theta}$

    To prove it, we will rederive the magnitude of the force by using the pythagorean theorem, that tells us that:

    $\displaystyle F_T^2 = F_x^2 + F_y^2$

    $\displaystyle F_T^2 = (F_T\cos{\theta})^2 + (F_T\sin{\theta})^2$

    $\displaystyle F_T^2 = F_T^2\cos^2{\theta} + F_T^2\sin^2{\theta}$

    We know that $\displaystyle \cos^2{\theta} + \sin^2{\theta} = 1$ so we get:

    $\displaystyle F_T^2 = F_T^2$

    $\displaystyle |F_T| = |F_T|$

    And there you go.

    The final statement is:

    $\displaystyle \vec{F} = F_x\hat{i} + F_y\hat{j}$
    Last edited by Aryth; Mar 26th 2009 at 10:35 AM.
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