# Are the ideas of angular velocity, angular acceleration, angular momentum, torque, etc only helpful for rigid bodies?

For rigid bodies, all the particles can have different linear velocities but the same angular velocity, so it makes it convenient to talk about the angular velocity instead. From there, we get to ideas like angular momentum and torque, which work the same way for angular motion as momentum and force do for linear motion.

However, if we have a system of $$n$$ particles freely moving around, say a gas, do we still use these ideas? In that case, the moment of inertia is constantly changing.

If we apply a constant force to any of the particles, then it'll result in a non-constant torque because of the continuously changing position vector. In case of rigid bodies, this is not the case because, in at least the 'axis of rotation frame', the torque due to a constant force on a particle is constant because the angle between the force and the position vector of the particle remains constant because of the rigid nature of the body.

For non-rigid bodies, there is no 'axis of rotation' frame, so torque is also very inconvenient to talk about.

So are these ideas only used for motions where the whole system can be ascribed the same angular velocity at all times?

Even though they may be different to think about than with rigid bodies, angular momentum is still a conserved quantity. Understanding the angular momentum of a gas cloud yields interesting constraints on how it can collapse into a black hole.

Torque and angular momentum don't depend on the system having a particular rotation. We can pick any axis we want and calculate torques and angular momentum from there.

• Wouldn't torque be very inconvenient to work with for gases? Even for a constant force applied on a gas particle for some duration, the torque would not be constant throughout the duration. Commented Aug 19, 2020 at 6:23
• Probably inconvenient. But it wouldn't be force on a particle, it would be force at a location. Individual articles can move out of the way as long as other particles move in. Commented Aug 19, 2020 at 15:40

Angular velocity and acceleration are only used in the context of rigid bodies, as far as I'm aware, because otherwise, you don't have a well defined axis of rotation. However, torque and angular momentum can be derived without reference to these quantities, and are used in more general situations. An axis of rotation is not needed to discuss torque, as it is primarily used in equations which do not refer to an axis of rotation: $$\vec \tau = \vec r \times \vec F$$ $$\vec \tau = \frac{\mathrm d \vec L}{\mathrm d t}$$ One example of where this is used outside of rigid body dynamics is in orbital dynamics, where you can use the fact the gravity does not apply any torque to planets to say the total angular momentum is conserved in orbits.

• I've seen this being applied to the orbital mechanics of Earth, but there there's only one particle in the system we're considering (i.e. Earth). So that's effectively just the special case of a rigid body (a rigid body having one particle) Commented Aug 19, 2020 at 6:32
• @RyderRude any orbit has two particle; for example, the Earth and the Sun. In this case, the Sun has a much larger mass than the Earth, so it is common to treat the Sun as a fixed point, but it is actually part of the orbit, and the same principles can be applied even for equal masses orbiting each other. Commented Aug 19, 2020 at 13:51

Yes. Concepts such as moment of inertia, torque, angular momentum etc are concepts which are applicable to rigid bodies. Unlike a collection of $$n$$ particles, as is the case for a gas, a rigid body is a solid body where deformation is zero or so small that it is negligible. Also, the distance between any two points on a rigid body is always constant (by definition) and so it would be meaningless to try to ascribe any quantities from rigid body dynamics to a gas since the distances between any two molecules is constantly changing.

• I think the ideas are still applicable for any system of particles (rigid nature of body is not an assumption while deriving these ideas). It's just that they might not be convenient to work with for gases. Commented Aug 19, 2020 at 4:53
• Angular momentum is definitely not exclusively applicable to rigid bodies. Commented Aug 19, 2020 at 5:15