# Does a general rotational analogue of CoM work exist?

• etotheipi
In summary, the conversation discusses the concept of center of mass work for general and rigid bodies and how to calculate the change in kinetic energy in the rest frame of the center of mass. It is mentioned that for rigid bodies, an expression for this change is easily found, but for non-rigid bodies, an integral over all point-masses is required. The conversation ends with the realization that the change in kinetic energy in the rest frame of the center of mass can be calculated using the WE theorem.
etotheipi
For a general body, there exists the notion of centre of mass work; that is, computing the work done if all of the forces on the body act through the centre of mass. If we separate the total kinetic energy into that of the CM and that relative to the CM, ##T = T_{CM} + T*##, we can show by integrating Newton's second law that if ##\vec{F}## is the resultant force, $$\int \vec{F} \cdot d\vec{r}_{CM} = \Delta T_{CM}$$Evidently since ##\Delta T = \Delta T_{CM} + \Delta T*##, it would be nice if we could also find an expression for ##\Delta T*##.

For rigid bodies, such an expression is not hard to find. If, in one dimension, ##\tau## is the total torque about the centre of mass, then we can integrate w.r.t. the angular position of the rigid body $$\int_{\theta_1}^{\theta_2} \tau d\theta = \int_{\theta_1}^{\theta_2} I_{CM} \frac{d\omega}{dt} d\theta = \Delta \left(\frac{1}{2} I_{CM} \omega^{2}\right) = \Delta T*$$ That's nice, but the general equation for CM-work ##\int \vec{F} \cdot d\vec{r}_{CM} = \Delta T_{CM}## also applies to bodies that are not rigid. But with the rotational version above, it is required that we know the angular displacement of the rigid body ##\theta_2 - \theta_1## and consequently we cannot use it for non-rigid bodies which do not have a well defined angular displacement.

So I wondered whether anyone knew of a form with which we can calculate ##\Delta T*##, without computing the real work ##\Delta T##. Thank you!

etotheipi said:
..., So I wondered whether anyone knew of a form with which we can calculate ΔT∗, without computing the real work ΔT. Thank you!
...
Are you asking about the change in kinetic energy in the rest frame of the center of mass? For the most general non-rigid body you have to integrate over all point-masses.

vanhees71 and etotheipi
A.T. said:
Are you asking about the change in kinetic energy in the rest frame of the center of mass? For the most general non-rigid body you have to integrate over all point-masses.

Yeah that's it. I wonder what such an integral would look like? Because if you use the construction $$W = \int \sum_i \tau_i d\theta_i = \Delta T$$ that gives you the total work (##\Delta T_{CM} + \Delta T*##) for general rigid body motion with torques calculated about a certain arbitrary coordinate system.

Is there any way of isolating the ##\Delta T*## in the rest frame of the centre of mass?

A.T. said:
For the most general non-rigid body you have to integrate over all point-masses.

Actually, I've woken up now and I understand. It's just an application of the WE theorem in the centre of mass frame. Gotcha. Thanks!

## 1. What is a general rotational analogue of CoM work?

A general rotational analogue of CoM work is a concept in physics that refers to the conservation of angular momentum in a rotating system. It states that the total angular momentum of a system remains constant unless acted upon by an external torque.

## 2. How does the general rotational analogue of CoM work differ from the traditional CoM work?

The traditional CoM work only applies to linear motion, while the general rotational analogue applies to rotational motion. The traditional CoM work is based on the conservation of linear momentum, while the general rotational analogue is based on the conservation of angular momentum.

## 3. Can you provide an example of the general rotational analogue of CoM work in action?

One example is a spinning top. As the top spins, its angular momentum remains constant, despite external forces such as gravity acting on it. This is due to the general rotational analogue of CoM work.

## 4. Is the general rotational analogue of CoM work a fundamental law of physics?

Yes, the general rotational analogue of CoM work is a fundamental law of physics that applies to all rotating systems. It is a consequence of the conservation of angular momentum, which is a fundamental principle in physics.

## 5. Are there any exceptions to the general rotational analogue of CoM work?

In certain cases, external torques can cause changes in the angular momentum of a system, violating the general rotational analogue of CoM work. This is known as torque-induced precession and is observed in systems such as gyroscopes and spinning tops.

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