Does rigid body rotation contradict Newton's first law?

Per Newton’s law, an object will move along a straight line with a constant speed if no force is acting upon it.

No portion of a rotating ball is moving in a straight line (except those on the axis which are static). How is it possible then that the ball will keep rotating in the absence of an external force?

• There are still molecular forces. If you magically remove them the object sheds and all the tiny pieces will go in a straight line. Commented May 13, 2023 at 16:26
• If the ball is made of something unstable like jelly, indeed you will find that rotating it fast enough, chunks of jelly will fly away in straight lines all over the place :) So the conclusion is, the molecules that a ball of stiff enough material is made of, are kept in place by a centripetal force which is responsible for maintaining them in this circular motion. In such a situation, another conservation law comes into play that Newton also derived: that of angular momentum under the influence of a central force.
– Amit
Commented May 13, 2023 at 16:27
• Another way to think about it is that the ball is at rest in a frame rotating with its angular speed around its axis. The lab frame is thus a non-inertial frame for the ball, giving rise to an acceleration, leading to forces
– Rol
Commented May 15, 2023 at 13:16
• @Quillo Indeed. Commented May 16, 2023 at 12:11
• @Amit: This exactly describes the principle of a spin dryer, where water molecules are made to fly off clothes by rotation. Commented May 16, 2023 at 13:44

First of all, a more precise statement is that "an object will move with constant velocity if there is no external force acting upon it". This is the same as saying the object moves in a straight line, unless its velocity is zero, in which case it doesn't move. It has already been pointed out in comments that individual pieces of the ball are not objects under no external force.

The usual statement that an object with no external force moves with constant velocity applies generally to the motion of its center of mass. Alternatively, it applies to point masses (for which it is hard to speak meaningfully of rotations), but this is just a special case of the previous statement. In your case, the center of mass does have a constant velocity: it is zero.

To describe the motion of a system more completely, we could say "the center of mass of a system with no external force or torque acting upon it has a constant velocity, and the system has constant angular momentum about its center of mass". This is admittedly not as short and sweet. For a rigid body, we could say "a rigid body moves with constant velocity while it rotates about its center of mass with constant angular momentum".

• The rotation of a rigid body is a very complex problem (in general it's chaotic). Your last statement, while it is trying to be helpful, is probably a little misleading. The best one can say is probably what you had before, that the angular momentum is constant. That we can express all rigid body rotation with constant angular momentum as the rotation around one axis is certainly not the case. Commented May 13, 2023 at 17:34
• @FlatterMann Yes, thank you. In general the angular velocity would only be constant if it aligns with a principal axis of inertia (even then the rotation about the axis with the intermediate moment of inertia isn't stable). Here is some further reading that might interest anyone looking for more on this.
– Puk
Commented May 13, 2023 at 17:48
• @KCd I didn't say or imply constant velocity means zero velocity.
– Puk
Commented May 15, 2023 at 0:03
• I believe this answer would be improved by removing the claim that "move with constant velocity" is more precise than "move along straight line with constant speed". The two statments are exactly equivalent, and both may be considered technically wrong in the case of zero velocity. However, the wording proposed in the answer is certainly more concise and conventional, so I agree it is worth proposing, minus the claim about precision. Commented May 15, 2023 at 11:19
• @Vaelus "move with constant velocity" is not technically wrong in case of zero velocity, unless you're being nitpicky about the use of the word "move"; on the other hand "move along a strait line" doesn't work as well if the "line" is actually a point. Commented May 16, 2023 at 17:39

If you look at the ball as a whole, it has no net force and moves in a straight line. However, if you just look at one portion of the ball, you must consider the forces which act on that portion.

Each portion of the ball (except those on the axis) is being pulled towards the axis by intermolecular forces and whatnot, so they do not continue moving in straight lines. That's because the intermolecular forces are trying to keep the ball in the same shape.

If those forces would go away, the portions of the ball would continue moving in straight lines. If you spin a CD very fast, those intermolecular forces aren't good enough and the pieces come apart and move in straight lines.

Per Newton’s law an object will move along straight line with constant speed if no force is acting upon it.

This statement (like all the Newton laws) applies to a point-like object. A rotating ball is clearly not viewed as a single point, but as a collection of points connected via links (a solid object or rigid body) - indeed, it is meaningless to talk about point-like object rotating about itself. These points do exert forces in each other.

Newton's 1st law states, as you quote, that an object in straight-line motion will seek to remain in straight-line motion.

But it does not state the opposite, that a an object in non-straight-line motion won't keep moving in that particular way.

To figure out if other types of motions also are kept, we must emply some other laws. It turns out that from Newton's 3rd law you can translate the tendency of straight-line motion to rotational motion in the case of rigid bodies, since intermolecular forces cause turning of each particle. Overall, pure turning with no translational effects take place, which we from the formula for centripetal acceleration know means that all particles within the rotating object will not slow down. Overall, we thus have rotation that isn't slowed down without any external torques acting.

To answer your question more clearly (or shall I say with more relative freedom) we can make a few basic assumptions about a particular scenario. These assumptions are not strictly necessary but they make the situation simple enough to only include basic physics

• Your ball is a rigid ball in an isolated stretch of spacetime, with no other forces (eg. gravity, magnetic fields) acting upon it.
• It is rotating slowly (meaning any point of the ball doesn't reach relativistic speeds)
• The ball is not a quantum scale object.
• The ball is sufficiently small enough so gravitational effects are negligible upon the rotation considered.
• We exclude other types of motion that can happen to the ball or its components like vibration

Then your ball isn't moving anywhere. It is rotating about a static axis in order to conserve its angular momentum. Yes, it has molecules moving in a circular path and the tendency of molecules to move away from the ball at tangential velocity is balanced out by intermolecular forces acting inward. So in the absence of net force centre of the ball keeps stillness (the centre is depicted as a point , and rotation of a point around its own axis is undefined because it is singular), and the molecules of the ball (or any point not in the exact centre of the ball) keep moving in a circular path at a constant angular velocity.

So in essence, while the ball doesn't have external forces acting upon it still has internal intermolecular (and also intra-molecular) forces acting upon every particle of it so as to convert the linear momentum of each particle into angular momentum every instance, hence linear velocity motion of particles are converted to circular motion. That is what rotation is. Every particle is subjected to the external force of other particles residing by it hence the circular motion (due to the gradient of linear velocity along the axis normal to the axis of rotation) of the particles of the rotating rigid body

Important question: Why is there a difference in linear velocity along the axis normal to the axis of rotation?

Answer: Because that is how rotational motion is imparted upon our rotating ball at the start of our motion, that is to make a positive gradient of linear velocity outward along the axis perpendicular to the axis of rotation. Physically an one way to do this is by applying two equivalent yet opposite forces at two exactly opposite points on a perimeter of a large circle on the ball.

Relevant question: What happens to the molecules at the exact axis of rotation?

Answer: Well, they rotate about the exact axis of the rotation of the ball. But still, parts of them (atoms) will be moving around (means in a circular path not changing the relative position of one point to another point of the ball). If you zoom in to more than that then there will be quantum effects and Heisenberg uncertainty comes into play.