- #1
FaraDazed
- 347
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Homework Statement
An object is rotating at 4 rad/s about an axis in direction of [itex](2 \hat{i} - 4 \hat{j} + 3\hat{k} ) [/itex] which then passes through a point (1,2,0)m .
Calculate the instantaneous velocity at the point (2,0,3)m (Hint: use [itex] \vec{v} = \vec{\omega} \times \vec{r} [/itex])
Homework Equations
[itex]
\hat{A} = \frac{\vec{A}}{|\vec{A}|}
[/itex]
The Attempt at a Solution
First off, i would like to say I am extremely new (1-2 weeks new) to cross products, and this is the first time we have been given physical (rather than purely mathematical) problems to implement it.
What I immediately thought of was the equation I put in the relevant equations section, how a unit vector is calculated and since all I need/want from the vector [itex](2 \hat{i} - 4 \hat{j} + 3\hat{k} ) [/itex] is the direction I turned that into a unit vector, and then to get the angular velocity vector times that by 4. So...
[itex]
\sqrt{2^2+4^2+3^2} = \sqrt{29} \\
\therefore \vec{\omega} = (\frac{4 \cdot 2}{\sqrt{29}} \hat{i} - \frac{4 \cdot 4}{\sqrt{29}} \hat{j} \frac{4 \cdot 3}{\sqrt{29}} \hat{k}) rad/s = (\frac{8}{\sqrt{29}} \hat{i} - \frac{16}{\sqrt{29}} \hat{j} + \frac{12}{\sqrt{29}} \hat{k}) rad/s[/itex]
And then converted to decimal to makes things easier
[itex]
\vec{\omega} = (1.486 \hat{i} - 2.971 \hat{j} + 0.557 \hat{k}) rad/s
[/itex]
And then to get [itex] \vec{r} [/itex] I did (1,2,0)-(2,0,3)=(-1,2,-3)
And then did the cross product, I am not sure how to do matrices in latex but I put i,j,k on top row (of 3 by 3 matrx) and then on second row put the omega vector and then on third put (-1,2,-3).
Then I got the determent and thus the velocity vector to be...
[itex]
\vec{v} = [(-2.971 \cdot -3)-(0.557 \cdot 2)] \hat{i} + [(1.486 \cdot -3)-(0.557 \cdot -1)] \hat{k} + [(1.486 \cdot 2)-(-2.971 \cdot -1)] \hat{k} \\
\vec{v} = 7.8 \hat{i} -3.9\hat{k}+0.001\hat{k}
[/itex]
I don't know if my method is correct at all, I am suspicious of the low value for the k component for a start off. Oh and as the question asks for "the instantaneous velocity" I do not know whether they mean just the velocity vector or its magnitude; I assume if they meant the magnitude then they would have just said "speed" instead, though.