Can a rotating object accelerate by changing shape?

Question

Suppose an object is already rotating in a situation of no external forces such as gravity or friction. Is it possible or impossible for it's velocity (linear) to change by the shape of the object changing? For example if a piece of it broke off then collided and stuck back to the object in a different spot. It might be hollow inside and the broken off piece traveled from one wall to the other wall of the hollow.

Answer 1

If not affected by an external force, the momentum of the object $\vec{p} = m\vec{v}$ is conserved. This means that it is not possible to accelerate the object by changing its shape. It does not matter if the object is rotating or not. However, if you permit the object to shed some of its mass the situation is very different; this is the working principle of a rocket, which accelerates by spewing out propellant at a high speed.

Edit: The only thing an object can achieve by changing shape by shifting around some of its parts (without shedding mass) is to increase the speed some of its parts while simultaneously decreasing the speed of other parts. This would happen in the situation you are proposing, where a broken piece travels from one wall to another. The speed that appears to be gained (by the walls) will then be lost again as the broken piece hits the opposite wall. Note also that in this case, the center-of-mass velocity of the object (consisting of walls + broken piece) is constant at all times.

Answer 2

It is kind of weird that you are asking it this way. Typically when people speak of rotating objects accelerating, they mean angular acceleration. The stereotypical example of this happening is the figure ice skater pulling their arms in to rotate faster.

This is possible because in the absence of an external torque, the angular momentum $L$ is conserved. Unlike linear momentum $p = m v$, of which the mass in this expression cannot be meaningfully changed by internal interactions, in the case of angular momentum, $L = I \omega$, and $I$ can be internally altered. Considering RKE $ = \frac12 L I^{-1} L$, this means that if you speed up the angular rotation by reducing $I$, you have to expend energy to do that. i.e. The figure skater would have to use stored food chemical potential energy in arm muscles to pull arms inwards in order to spin faster.

So no, linear acceleration of the centre of mass cannot happen that way.

Answer 3

It is not possible in ordinary flat space. It might be possible in a curved space.

Consider compact 2D shapes sliding around on the frictionless surface of a large sphere. The possible motions are rotations. We can represent them as rotations about three orthogonal axes, one of them chosen to be through the middle of the object, and the other two far away. If the sphere is big enough, it looks like a flat plane to an observer looking at the local picture, and now rotations about the two distant axes look approximately like 2D translations in a plane, and the third axis looks like rotation in the plane. We can use this to draw an analogy between rotational motion on the sphere, and Euclidean motion (rotation+translation) on the plane. The angular momentum about the two distant axes turns into linear momentum as the distant axes head off to infinity. The moment of inertia about the two distant axes turns into the mass of the object.

This connection between angular and linear versions motion/momentum/mass can be made explicit by embedding Newtonian physics into projective geometry. For motion in 3D we add another direction (let's call it $w$) so we have a 4D space (and we position our flat 3D universe on the hyperplane with $w=1$). Then rotations have six degrees of freedom (one for every pair of axes: $wx$, $wy$, $wz$, $yz$, $xz$ and $xy$), and three of them head off to infinity and become the mass (the ones with a $w$ in them), and the other three stay behind as the moment of inertia. Linear and angular momentum are likewise just different components of the same six-dimensional quantity, as are forces and torques.

As others have noted, in flat space the mass can not change (the distance to the distant axes of rotation is always the same: infinity), and so the position of the centre of mass is fixed. But space is not necessarily flat. And if it is not, we can change the mass like we can change the moment of inertia - by changing shape.

So imagine we have a stationary blob of matter sat on the equator of our sphere. We split it into two equal sized blobs connected by a thin thread, push the blobs away to the two poles of the sphere, now move the thread around the sphere, which is easy because it is so thin and light, then draw the blobs back together to a new place on the equator. We can move the blob to anywhere, without any net angular momentum. If we repeatedly cycle small moves, we can make the blob appear to 'translate' across the sphere just by changing shape. In general, we don't have to go all the way to the poles - we only have to be able to spread out over an appreciable fraction of the 'radius of curvature' of space in your vicinity in order to change the mass/moment of inertia, which makes reactionless 'linear' motion possible in curved space.

However, since the radius of curvature of the universe has been found to be far larger than the observable part of it (14 billion light years), this isn't really a practical notion for long-range travel!