Register to reply

Cross product and dot product of forces expressed as complex numbers

Share this thread:
magwas
#1
Feb11-10, 05:10 AM
P: 125
1. The problem statement, all variables and given/known data

I have came up with an example to illustrate my question.

There is a rod, which can turn around p1.



p1p2 = (-1+j) m
p1p3 = (-3 + 3j) m
p1p4 = (1 - j ) m
F1 = (1+3j) N
F3 = (-1 - 2j ) N
F4 = unknown, orthogonal to the rod

compute F2_n, orthogonal component of F2 to the rod
compute F2_t, paralell component of F2 to the rod


2. Relevant equations

The question is actually here:
The sum of moments is
[tex]\sum{\vec{F} \times \vec{l}} =0[/tex]
Where
[tex]a \times b = \Re{a} \Im{b} - \Im{a} \Re{b}[/tex]
Is that true?
Likewise, the force components paralell to the rod is:
[tex]\sum{\vec{F} \cdot \hat{\vec{l}}} = 0[/tex]
where
[tex] a \cdot b = a \overline{b} + b \overline{a} = 2 \Im{a} \Im{b} + 2 \Re{a} \Re{b}[/tex]
Is it correct?
3. The attempt at a solution

I write the moments around p3. I sum here because:
  • all forces are on the same side of the turning point
  • all arms are measured towards the turning point (this is why p1p3 - p1p4)
  • the direction of forces are encoded in their vectors
The unit vector normal to the rod is come by dividing a vector along the rod by its length, and multiplying it with j: [tex]\frac{\mathbf{\imath} p1p3}{\lvert{p1p3}\rvert} [/tex]
so the equation for moments:
[tex] F_{1} \times \left(p1p3 - p1p4\right) + F_{3} \times \left(p1p3 - p1p2\right) + p1p3 \times \left \frac{\mathbf{\imath} p1p3}{\lvert{p1p3}\rvert} \lvert F_{2_{n}}\rvert} = $\\
$
\Im{p1p3} \Im\left(\frac{\lvert F_{2_{n}}\rvert p1p3}{\lvert{p1p3}\rvert}\right) + \Im\left(p1p3 - p1p2\right)
\Re{F_{3}} + \Im\left(p1p3 - p1p4\right) \Re{F_{1}} + \Re{p1p3} \Re
\left(\frac{\lvert F_{2_{n}}\rvert p1p3}{\lvert{p1p3}\rvert}\right) - \Im{F_{1}} \Re\left(p1p3 - p1p4\right) -
\Im{F_{3}} \Re\left(p1p3 - p1p2\right) = $\\
$
10.0 + 4.24264068711929 \lvert F_{2_{n}} \rvert = 0[/tex]
so
[tex]\lvert F_{2_{n}}\rvert =-2.3570226039551 [/tex] which gives
[tex]F_{2_{n}} = \lvert F_{2_{n}}\rvert \frac{\mathbf{\imath} p1p3}{\lvert{p1p3}\rvert} = 1.66666666666667 + 1.66666666666667 \mathbf{\imath}[/tex]

Now the forces paralell to the rod:

We use our unit vector [tex]\hat{l} = \frac{p1p3}{\lvert{p1p3}\rvert}[/tex]
, and forget F4 as it is orthogonal to the rod, so the sum:
[tex] F_{3} \cdot \hat{l} + \lvert F_{2_{t}}\rvert \cdot \hat{l} + F_{1} \cdot \hat{l} = $\\
2 \lvert F_{2_{t}}\rvert \Re{\hat{l}} + 2 \Im{F_{1}} \Im{\hat{l}} + 2 \Im{F_{3}} \Im{\hat{l}} +
2 \Re{F_{1}} \Re{\hat{l}} + 2 \Re{F_{3}} \Re{\hat{l}} = $\\
1.4142135623731 - 1.4142135623731 \lvert F_{2_{t}}\rvert = 0 [/tex]
so
[tex]\lvert F_{2_{t}}\rvert = 1[/tex]
which gives
[tex] F_{2_{t}} = -0.707106781186548 + 0.707106781186548 \mathbf{\imath} [/tex]
and
[tex] F_{2} = F_{2_{n}} + F{2_{t}} = 0.959559885480119 + 2.37377344785321 \mathbf{\imath}[/tex]
Attached Thumbnails
example.png  
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
magwas
#2
Feb11-10, 05:14 AM
P: 125
Well, maybe I should have used [tex]magnitude_{F_{2_{n}}}[/tex] instead of [tex]\lvert F_{2_{n}}\rvert[/tex]...
nvn
#3
Feb11-10, 10:40 AM
Sci Advisor
HW Helper
P: 2,124
magwas: I got F2n = 2.3570 N, but I got F2t = 0.707 107 N, not 1. You can check your answer by summing forces in the rod tangential direction, to see if the summation equals zero.

magwas
#4
Feb11-10, 12:51 PM
P: 125
Cross product and dot product of forces expressed as complex numbers

I see, [tex]\lvert F_{2_{t}}\rvert \cdot \hat{l} [/tex] was a mistake.
the equation correctly is [tex]F2t + \left ( F_{1} \cdot l \right) + \left ( F_{3} \cdot l \right) = 0[/tex]
but it comes down to
[tex]F2t + 2 \Im{F_{1}} \Im{l} + 2 \Im{F_{3}} \Im{l} + 2 \Re{F_{1}} \Re{l} + 2 \Re{F_{3}} \Re{l} = 0[/tex]
which leads to [tex]1.4142135623731 + F2t = 0[/tex],
so F2t = -1.4142135623731
Do I have a problem with the definition of complex dot product?

Thank you again.
magwas
#5
Feb11-10, 05:32 PM
P: 125
I have looked up the definition of vector dot product. Wikipedia tells me that it is
[tex]\sum a_{i} b_{i}[/tex] for vectors a=(a1,...,an) and b=(b1,...bn).

So a . b must be re(a)re(b)+im(a)im(b), not twice that.
In this way I get the same result as you, I believe.


Register to reply

Related Discussions
Inner Product of 0 vector, & Complex numbers Linear & Abstract Algebra 3
Cross product forces Classical Physics 13
What is cross product of two complex numbers? Calculus & Beyond Homework 31
Cross Product, Forces Introductory Physics Homework 10
Complex Cross Product and Area of a parallelogram Calculus & Beyond Homework 3