Do objects with non-uniform shapes and mass distributions rotate as a result of gravitational attraction?

Question

I was thinking about orbital mechanics this morning and a question arose: do objects with non-uniform shapes and mass distributions rotate as a result of gravitational attraction?

Thinking through the problem, I know that two point masses are attracted by a force based on Newton's universal gravitational equation:

$$F=\frac{G{m_1}{m_2}}{r^2}$$

To keep things simple, let's make one of the masses a point mass, $m_p$, and have just one of the objects be non-uniform.

If we subdivide the non-uniform object into voxels and represent each voxel as a point mass, we could then have $M$ be the set of masses and $R$ be the set of distances from the point mass $m_p$. The gravitational force experienced by each voxel would then be:

$$F_i = \frac{Gm_pM_i}{{R_i}^2}$$

Using $F=ma$, the acceleration experienced by each voxel would then be:

$$a_i = \frac{Gm_pM_i}{{R_i}^2M_i} = \frac{Gm_p}{{R_i}^2}$$

This was briefly surprising until I remembered that heavier objects don't fall faster than lighter objects in a vacuum, so the voxel acceleration is independent of the voxel mass. However, this still leaves distance as a variable.

Intuitively, though, if one point on a rigid body is accelerating faster than another point on the same rigid body, it feels like it must be changing its orientation.

This is where I got stuck. I tried thinking about this in terms of torque, trying to resolve the angular acceleration experienced around the center of mass based on the sum of the $\tau=r \times F$ dot product vectors, but I got stuck trying to figure out the moments of inertia.

I considered that there may be some sort of additional transference of force involved due to the rigid body constraint, essentially causing the force felt by the far-side voxel to be also applied to the near-side voxel, akin to the normal force, but that seems a bit wishy-washy considering that you can apply a linear force to the outside edge of a wheel and cause it to spin. I've got a feeling that this might be fallacious thinking due to the angle of the applied force vector mattering, but I'm not 100% sure.

So, what's the deal? Do objects rotate due to the force of gravity?

(Note: I would particularly appreciate answers that offer intuitive explanations alongside the mathematics, since I'm a bit rusty!)

Answer 1

Gravity can make things rotate, sure. Here is a simple example.

The object we are interested in is the balanced rigid dumbbell on the left. The rod connecting the two masses at the ends has negligible mass. Suppose everything is initially at rest. The mass above is closer to the mass $M$ to its right, so it gets pulled harder than the mass below. Thus there is a net clockwise torque about the center of mass of the dumbbell, causing it to start to rotate clockwise.

The fact that gravity can make things rotate doesn't mean angular momentum isn't conserved. The mass to the right begins to accelerate roughly toward $m_1$, gaining a counterclockwise angular momentum about the system's center of mass that balances that of the dumbbell.

Answer 2

Intuitively, though, if one point on a rigid body is accelerating faster than another point on the same rigid body, it feels like it must be changing its orientation.

The problem is, you are only considering the force due to gravity. You also have the forces between the atoms in the material that hold it together. At least for everyday objects, these forces are much, much stronger than the gravitational force.

So when you write $$F = ma,$$ $F$ is actually the sum of all forces that act on your Voxel. If you consider a rigid object, the sum of all forces (Bonding force + gravitational Force) will amount to zero.

Now, if you consider an object that is actually a swarm of particles that only interact through gravity, then your reasoning would be somewhat correct. (Think of a galaxy, where the individual particles are the stars). But it's the other way around. Gravity pulls every particle into the center, so if you want the object to be stable and not collapse, you have to give it some rotation.

Gravity is not causing the object to rotate; the shape of the object is the result of rotation.

Answer 3

Yes. Gravity gradients (tides) can cause objects to rotate. It's a very significant effect for spacecraft in low Earth orbit. Compensating for gravity gradient torques is a major problem for spacecraft attitude control.

For natural bodies, an extreme example is chaotic rotation.