Posted by sam on Wednesday, March 28, 2007 at 7:32pm.
In unit-vector notation, what is the net torque about the origin on a flea located at coordinates (-2.0, 4.0 m, -1.0 m) when forces F1 = (-4.0 N) k and F2 = (-5.0 N) j act on the flea?
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For Further Reading
* math (vectors) & physics - drwls, Thursday, March 29, 2007 at 12:13am
Torque is the cross product of the radius vector and the net force vector. In three dimensions, it can be computed as fallows.
|i... j... k|
|0... -5... -4|
|-2... 4... -1|
Compute the determinant
~~~
this is what i've done and im stuck.
= i[(-5)(-1)-(-4)(4)] + j[(-4)(-2)-(0)(-1)] + k[(0)(4)-(-5)(-2)]
= 21i + 8j - 10k
and that's the wrong answer.. .what am i doing wrong? help?
***there's an example in my book of a similar problem with different numbers:
-In unit-vector notation, what is the net torque about the origin on a flea located at coordinates (0, –4.0 m, 5.0 m) when forces F1=(3.0 N)k and F2=(-2.0 N)j act on the flea?
~~~~~and the answer to that question is: (–2.0 i) N · m
I agree wth your numbers. However if the right hand rule is used for torque, then it should have been F x R instead of R X F. This would change the sign of the answer.
To find the net torque about the origin, you need to calculate the cross product of the radius vector and the net force vector. In this case, the radius vector is the position vector of the flea, and the net force vector is the sum of forces F1 and F2.
Let's denote the position vector of the flea as R = (-2.0, 4.0, -1.0) and the forces F1 and F2 as F1 = (-4.0, 0, 0) and F2 = (0, -5.0, 0) respectively. Notice that F1 only has a non-zero component in the k-direction, while F2 only has a non-zero component in the j-direction.
Now, to calculate the cross product, we can use the determinant method. Consider the following matrix:
|i j k|
|0 -5 -4|
|-2 4 -1|
Compute the determinant:
= i[(-5)(-1)-(-4)(4)] + j[(-4)(-2)-(0)(-1)] + k[(0)(4)-(-5)(-2)]
= i[5 - 16] + j[8 - 0] + k[0 - 10]
= i[-11] + j[8] + k[-10]
Therefore, the net torque about the origin on the flea is given by (21i + 8j - 10k) N·m.
However, based on your book's example, it seems there might be a difference in the convention used for the cross product. If the right-hand rule convention is used, the cross product should be F x R instead of R x F. This would change the sign of the answer.
So, if we use the right-hand rule convention, the net torque about the origin on the flea would be (-21i - 8j + 10k) N·m.
Make sure to double-check the convention used in your book or by your instructor to ensure you apply the correct sign convention for the cross product.