# Understanding conservation of angular momentum in relation with rotating objects

Conservation of angular momentum says that the angular momentum of a closed system will not change if there is no external torque applied to the system. For example, let's take the example of a simplified/idealized Earth-Sun system, where the Earth is undergoing circular motion around the Sun. The angular momentum is conserved since there is no external torque applied. However, there is an internal force which is causing Earth to orbit the Sun, namely the gravitational force acting as centripetal force. If we would remove the centripetal force, Earth would leave the orbit around the Sun. This makes sense, because in order to have circular motion, we need a constant acceleration to change its direction.

However, if in space I would give a ball a spin and then release it, the ball would keep spinning due to the conservation of angular momentum. But the ball keeps spinning without any internal forces. There is no centripetal force acting on the ball that keeps it in circular motion, which does not make sense to me, because circular motion implies a constant acceleration to change its direction. How does this make sense? In both cases the angular momentum appears to be conserved, but with the Earth, there is a clear centripetal force keeping it in orbit, while with the ball, this is not the case.

The same with the internal spin of the Earth. The Earth is spinning due to the conservation of angular momentum around its own axis, but not as a result of a centripetal force like the gravitational force which causes the Earth to orbit the Sun.

In short, I do not understand how conservation of angular momentum is possible without an internal centripetal force, such as the gravitational force which keeps the Earth in orbit around the Sun. Circular motion in my view needs to have a force since it requires a constant acceleration to change its direction.

There is most certainly an internal centripetal force within the material keeping it in circular motion. In real objects, this is provided by the intermolecular/interatomic forces, which are never fully rigid. This is why the Earth has an equatorial bulge arising from its rotation. Without these forces holding everything in place, the object will cease to exist as a single object. In idealized rigid bodies, the distance between any two points remains constant (which is what it means to be a rigid body), so the internal forces will be whatever is needed to maintain the constant distances, much like forces of constraint.

Since the internal forces are internal, they do not apply any external torque. Angular momentum will be conserved regardless of whether these forces are present.

• I see, so there is still an internal force which causes Earth to spin. But then, this internal force should be considered external according to the frame of a human that is sitting on Earth right? Why do we then not feel this internal force, i.e. constant acceleration that keeps the Earth spinning? It is true that we are 'moving' along the Earth at an instantaneous moment in time, but then the Earth is changing its direction, so there should be a force acting on us to change our direction along with the Earth, which we should be able to feel. Commented Apr 14 at 10:57
• @Stallmp The internal forces do not cause Earth to spin. They hold it together. As for the centripetal force due to Earth's rotation, it is very small due to the rotation rate. The "gravitational force" dominates. Commented Apr 14 at 11:08
• @Stallmp The force you feel is the normal force applied by whatever surface is supporting your body. And this normal force is the difference between gravity and centripetal acceleration, which is less than half a percent of gravity, even at the equator. So you still feel more than 99.5% of gravity. Commented Apr 14 at 11:19
• @Stallmp That acceleration is the inward centripetal acceleration. But this centripetal acceleration does not in itself change the angular velocity as being a radial force, it applies no torque and does no work. Commented Apr 14 at 11:27
• FWIW, the centripetal acceleration on a body at the equator is ~0.0339 m/s^2, so it reduces the effective gravity by around 1 part in 290. Commented Apr 14 at 11:31

There is no centripetal force acting on the ball that keeps it in circular motion

This is not correct. Each part of the ball is held in place because it is connected to the surrounding parts of the ball. These internal forces make the ball rigid, and also provide the centripetal force on each part of the ball which allows the ball to spin as a rigid body. If the internal forces are too weak or the ball is spinning too quickly then it will fly apart.