Torque as measured from center of mass.

In summary, the total angular momentum of a system of particles can be separated into two components: the orbital angular momentum of the center of mass and the spin angular momentum with respect to the center of mass. The net external torque on the system can be expressed as a sum of two terms, one involving the center of mass and the other involving the individual particles. The change in orbital angular momentum is given by the cross product of the center of mass vector and the net force on the system, while the change in spin angular momentum can be nonzero even if the net force is zero.
  • #1
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The angular momentum of a system of particles α = 1,2,3,...n can be written as the sum of the two angular momenta:
[tex]\vec{L} = \vec{R} \times \vec{P} + \sum_{\alpha} \vec{r_{\alpha}}' x \vec{\rho_{\alpha}}'[/tex]
where the first term is the angular momentum of the center of mass with all mass M = Ʃ mα, and the second term is the total angular momentum of the system with respect to the center of mass.

If we differentiate with respect to time, we get the torque:
[tex]\vec{N} = \frac{d}{dt} \vec{L} = \frac{d}{dt} \vec{R} \times \vec{P} + \frac{d}{dt} \sum_{\alpha} \vec{r_{\alpha}}' \times \vec{\rho_{\alpha}}' = \vec{R} \times \vec{F} + \sum_{\alpha} \vec{r_{\alpha}}' \times \vec{F_{\alpha}}'[/tex]
where F is the net external force on the system and Fα' is the "effective" force as seen with respect to the center of mass.

We can write the effective force in terms of the real force by:
[tex]\vec{F_{\alpha}}' = \vec{F_{\alpha}} - m_{\alpha} \frac{d^2}{dt^2} \vec{R}[/tex]

And so the above becomes:
[tex]\vec{N} = \vec{R} \times \vec{F} + \sum_{\alpha} \vec{r_{\alpha}}' \times \vec{F_{\alpha}} - \sum{m_{\alpha} \vec{r_{\alpha}}' \times \frac{d^2}{dt^2} \vec{R}}[/tex]
But the second sum is zero because of how the center of mass is defined and so:
[tex]\vec{N} = \vec{N} = \vec{R} \times \vec{F} + \sum_{\alpha} \vec{r_{\alpha}}' \times \vec{F_{\alpha}}[/tex]

This says that:
[tex]\vec{R} \times \vec{F} + \sum_{\alpha} \vec{r_{\alpha}}' \times \vec{F_{\alpha}}' = \vec{N} = \vec{R} \times \vec{F} + \sum_{\alpha} \vec{r_{\alpha}}' \times \vec{F_{\alpha}}[/tex]

So eiher I have an inconsistency in my derivation or the torque can be measured by looking at the observed accelerations in both the inertial frame and the non-inertial frame so as long as the non-inertial frame has the motion characteristics as the center of mass. Is this true?
 
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  • #2
As you say, the total angular momentum separates into two parts: the one of the center of mass (orbital), and the one with respect to the center of mass (spin).
The time derivative of the total angular momentum will equal the net external torque on the system, which separates again into a term on the center of mass and a term with respect to the center of mass.

Some of the torque will contribute to the change in orbital angular momentum, and this is expressed just as the cross product of the center of mass vector and the net force on the total system.

The rest of the torque will contribute to a change in the spin angular momentum, and can be nonzero even if the net force is zero.
 

1. What is torque as measured from center of mass?

Torque as measured from center of mass is a measurement of the rotational force applied to an object, taking into account its mass and the distance from its center of mass to the point of rotation. It is a key concept in rotational dynamics and is often used to describe the stability and movement of objects.

2. How is torque as measured from center of mass different from regular torque?

Regular torque is measured from a fixed point of rotation, such as an axis or pivot point. Torque as measured from center of mass takes into account the object's center of mass, which may not always align with the point of rotation. This allows for a more accurate calculation of the rotational forces acting on an object.

3. Why is it important to consider torque as measured from center of mass?

Considering torque from the center of mass allows for a more accurate prediction of an object's rotational movement and stability. It takes into account the distribution of mass within an object and can help identify the point at which an object will balance or rotate.

4. How is torque as measured from center of mass calculated?

To calculate torque as measured from center of mass, you must first determine the distance from the center of mass to the point of rotation. Then, multiply this distance by the force applied perpendicular to this distance. This will give you the torque as measured from center of mass in units of newton-meters (Nm).

5. Can torque as measured from center of mass be negative?

Yes, torque as measured from center of mass can be negative. A negative torque indicates that the rotational force is acting in the opposite direction of the positive (counterclockwise) direction. This is often seen in situations where the force applied is closer to the center of mass than the point of rotation.

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