Calculating resultant torque using cross product

In summary: So if the Z component is 0, it means that the resulting vector is parallel to the XY plane, so the Z component is 0. The i and j components will depend on the values of the other two vectors.
  • #1
steffercakes
3
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1."In this exercise, you will be finding the resultant torque from the cross product of a lever arm with a force vector. The lever arm vector is A=2.0i+3.0j. The force vector is B=3.0i-4.0j.
Find A x B
B x A
and 2A x 3B




2.My teacher has been sick the past few days so hasnt taught us anything about torque yet, but the homework is still due. I'm not sure where to start because every formula I've read about how to do these says to include sinθ, but an angle was not given.



3. is it just: 6i-12j+1k
-6i+12j-1k
and 36i-72j+6k? Someone please help me understand how these are done.
 
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  • #4
steffercakes said:
1."In this exercise, you will be finding the resultant torque from the cross product of a lever arm with a force vector. The lever arm vector is A=2.0i+3.0j. The force vector is B=3.0i-4.0j.
Find A x B
B x A
and 2A x 3B




2.My teacher has been sick the past few days so hasnt taught us anything about torque yet, but the homework is still due. I'm not sure where to start because every formula I've read about how to do these says to include sinθ, but an angle was not given.



3. is it just: 6i-12j+1k
-6i+12j-1k
and 36i-72j+6k? Someone please help me understand how these are done.

There are two interpretations of the cross-product:

Geometrical Interpretation:

[itex] \vec{A}x\vec{B} = |A||B|\sin \theta \hat{n} [/itex]

Algebraic Interpretation:

[itex] \vec{A} = A_x i + A_y j + A_z k [/itex]
[itex] \vec{B} = B_x i + B_y j + B_z k [/itex]
[itex] \vec{A}x\vec{B} = (A_y B_z-A_z B_y)i + (A_z B_x - A_x B_z)j + (A_x B_y - A_y B_x)k [/itex]
 
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  • #5


Or you can learn from this video also.
 
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  • #6
CKOMETTER said:
There are two interpretations of the cross-product:

Geometrical Interpretation:

[itex] \vec{A}x\vec{B} = |A||B|\sin \theta \hat{n} [/itex]

Algebraic Interpretation:

[itex] \vec{A} = A_x i + A_y j + A_z k [/itex]
[itex] \vec{B} = B_x i + B_y j + B_z k [/itex]
[itex] \vec{A}x\vec{B} = (A_y B_z-A_z B_y)i + (A_z B_x - A_x B_z)j + (A_x B_y - A_y B_x)k [/itex]

ohhh okay, I see now. and if there's no Z (z=0), that makes that direction 0? So in this case the i and j are 0?
 
  • #7
steffercakes said:
ohhh okay, I see now. and if there's no Z (z=0), that makes that direction 0? So in this case the i and j are 0?

Yes, the cross-product of two vectors results in a vector that it's perpendicular to the plane that contains both vectors: in this case the Z direction.
 

1. What is the formula for calculating resultant torque using cross product?

The formula for calculating resultant torque using cross product is T = r x F, where T is the resultant torque, r is the position vector from the axis of rotation to the point of application of the force, and F is the force vector.

2. How do I determine the direction of the resultant torque?

The direction of the resultant torque can be determined by using the right-hand rule. Point your fingers in the direction of the force and curl them towards the position vector. The resultant torque will be in the direction perpendicular to both the force and the position vector.

3. Can the magnitude of the resultant torque be negative?

Yes, the magnitude of the resultant torque can be negative. This means that the direction of the resultant torque is opposite to the direction determined by the right-hand rule. It indicates that the force is trying to rotate the object in the opposite direction.

4. How does the angle between the force and position vector affect the resultant torque?

The angle between the force and position vector affects the magnitude of the resultant torque. When the angle is 0 degrees, the resultant torque is 0 as well. As the angle increases, the magnitude of the resultant torque also increases, reaching its maximum when the angle is 90 degrees. After that, the magnitude decreases as the angle approaches 180 degrees.

5. Can I use cross product to calculate resultant torque for any type of force?

Yes, cross product can be used to calculate resultant torque for any type of force, as long as the force and position vector are perpendicular to each other. This includes both linear and rotational forces.

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