Position vector in torque (and angular momentum)

In summary, torque is the cross product of position vector and force, and its magnitude is dependent on the choice of origin. This means that torque is not an absolute concept, and the same applies to angular momentum. In a general case, the torque at a given time is calculated as \vec r(t) \times \vec F(t) or (\vec r(t) - \vec r(t_1)) \times \vec F(t), and the angular momentum can be found with respect to any chosen origin.
  • #1
chuchung712
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Torque is defined as the cross product of position vector and force, i.e. [itex]\vec \tau = \vec r \times \vec F[/itex] .

However the force vector [itex]\vec F[/itex] is fixed, but the choice of origin is arbitrary, making [itex]\vec r[/itex] also arbitrary. Does it make the torque vector also arbitrary, which apparently shouldn't be?

So let's say in a very general case, a force [itex]\vec F(t)[/itex] acts on a particle between times t1 and t2 with position vector of the particle [itex]\vec r(t)[/itex]. Is the torque at time t simply [tex]\vec r(t) \times \vec F(t)[/tex] or [tex](\vec r(t) - \vec r(t_1)) \times \vec F(t)[/tex]? And if I want to find the angular momentum, is the linear momentum [itex]\vec p(t)[/itex] or [itex]\vec p(t_1) - \vec p(t)[/itex]? How do you justify the choice of origin as the centre of rotation of most standard cases? (as in https://commons.wikimedia.org/wiki/File:Angular_momentum_circle.svg)

Please correct me if I have any conceptual problems, but I am really confused.
 
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  • #2
Yes, the origin is arbitrary thus the torque. Whatever origin you define, you will find the angular momentum with respect to that point, so angular momentum is also not an absolute concept.
 
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  • #3
TESL@ said:
Yes, the origin is arbitrary thus the torque. Whatever origin you define, you will find the angular momentum with respect to that point, so angular momentum is also not an absolute concept.
Thank you so much. So in my case the torque is simply [itex]\vec r(t) \times \vec F(t)[/itex], right?
 
  • #4
Yes.
 

1. What is a position vector?

A position vector is a mathematical representation of a point or location in space. It is typically denoted by the symbol r and is defined as the displacement from the origin of a coordinate system to a particular point.

2. How is a position vector used in torque calculations?

In torque calculations, the position vector is used to determine the distance between the point of rotation and the point where the force is applied. This distance, along with the magnitude and direction of the force, is used to calculate the torque exerted on an object.

3. What is the relationship between position vector and angular momentum?

The position vector and angular momentum are directly related. Angular momentum is defined as the product of the moment of inertia and the angular velocity, and the moment of inertia is calculated using the position vector. Therefore, any changes in the position vector will affect the angular momentum of an object.

4. Can the position vector and torque be negative?

Yes, both the position vector and torque can be negative. A negative position vector indicates that the point is in the opposite direction of the origin, while a negative torque indicates a clockwise rotation.

5. How does the direction of the position vector affect torque?

The direction of the position vector relative to the direction of the force will determine the direction of the torque. If the position vector and force are parallel, the torque will be zero. If they are perpendicular, the torque will be maximized, and the direction of the torque will be determined by the right-hand rule.

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