# Can there be any change in angular velocity without internal torque if angular momentum is to be conserved?

Also, why is angular momentum defined only for one body but linear momentum for a system of bodies?

Coming back to my first question, angular momentum is conserved when no external torque acts on it. However, for a change in angular velocity, we need change in acceleration, which implies the need for torque. This torque can't be external if L is to be conserved. Is it then the internal torque? Can the velocity change without ANY torque?

Angular velocity is changed without any external torque when skaters extend or draw in their arms. Moment of inertia also changes as shape changes, but the product of the two, angular momentum, is conserved.

if, instead of a skater, you consider a pair of rotating weights held together by a wet string, as the string shrinks their rotation will speed up with no torque applied.

Angular momentum is defined for a system of bodies but you must nominate an axis for it.

• But there's some force acting on the skater and thus, internal torque? This is where I'm confused. Is it like in the case of linear momentum where internal forces do not affect momentum of the system? (Internal torque in this case) Commented Dec 25, 2019 at 16:49
• That is why I introduced the two weights model - to avoid the complexities of the arm joints of the skater. The model of two weights attached by a shrinking string has no internal torque.
– DrC
Commented Dec 26, 2019 at 0:48

Yes, you can have changes to rotational velocity with no external torque if the object is not rotating about any of the three principal axes of rotation.

One example is the Dzhanibekov effect where a free object (in space) changes orientation completely and suddenly all on its own.

Also, for a system of rigid bodies, you can indeed sum up with the total rotational momentum just as you sum up the total translational momentum

\begin{aligned} \boldsymbol{p}_{\rm total} & = \sum_i \left( m_i \boldsymbol{v}_i \right) \\ \boldsymbol{L}_{\rm total} & = \sum_i \left( \mathcal{I}_i \boldsymbol{\omega}_i + \boldsymbol{r}_i \times (m_i \boldsymbol{v}_i) \right) \end{aligned}

where $$m_i$$ is the mass of each object, and $$\mathcal{I}_i$$ is the mass moment of inertia tensor of each object (rotated to the common inertial frame).