Deformation of a spinning sphere

In summary: The stress in the ring is just the average stress over all the elements. This problem is called a Hooke's spring problem. You can find the solution by using ordinary differential equations.
  • #1
quantumfoam
133
4

Homework Statement

Hello guys! Now this is not a homework question, but it may sound like one. If a uniform sized and massed sphere was spinning in space away from any source of forces that could affect it, wouldn't the only forces that act on it are the centripetal and centrifugal? And if this sphere had elasticity, wouldn't the force that is stretching the sphere be either centripetal of centrifugal force? Now I am not including its gravitational force on purpose. Its gravitational field can be negligible. I also know that the centrifugal force is not a real force, but it is useful for explaining some phenomena. I'm using classical mechanics. The sphere will not be spinning at relativistic speeds.




Homework Equations



mωv=-kx

The Attempt at a Solution

Well, mathematically, I figured that if the centripetal and centrifugal forces are the sources of deformation of stretching of the sphere, then by using Hooke's Law we find that
mωv=-kx
where mωv is the centripetal or centrifugal force magnitude and -kx is Hooke's Law. Is this right? Or am I forgetting something? I would really appreciate anyone's help. My extremely inferior mind is not sure of itself.:frown:
 
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  • #2
quantumfoam said:

Homework Statement

Hello guys! Now this is not a homework question, but it may sound like one. If a uniform sized and massed sphere was spinning in space away from any source of forces that could affect it, wouldn't the only forces that act on it are the centripetal and centrifugal? And if this sphere had elasticity, wouldn't the force that is stretching the sphere be either centripetal of centrifugal force? Now I am not including its gravitational force on purpose. Its gravitational field can be negligible. I also know that the centrifugal force is not a real force, but it is useful for explaining some phenomena. I'm using classical mechanics. The sphere will not be spinning at relativistic speeds.




Homework Equations



mωv=-kx

The Attempt at a Solution

Well, mathematically, I figured that if the centripetal and centrifugal forces are the sources of deformation of stretching of the sphere, then by using Hooke's Law we find that
mωv=-kx
where mωv is the centripetal or centrifugal force magnitude and -kx is Hooke's Law. Is this right? Or am I forgetting something? I would really appreciate anyone's help. My extremely inferior mind is not sure of itself.:frown:

Yes, this problem can be solved, but not the way you suggested. This is a complicated problem in Theory of Elasticity. You need to use Hooke's Law in its full tensorial form to get the stresses that develop in the sphere. This includes hoop stresses that are important in supporting the centrifugal acceleration. The stresses that develop have to combine properly with the centrifugal forces. The boundary conditions involve the stresses at the surface of the sphere: the stress tensor dotted with a unit normal to the surface must be equal to zero. This is a 2D problem for the latitudinal and radial displacements; the longitudinal displacement is equal to zero. It's an interesting problem, but takes some work to solve.
 
  • #3
Oh I know I could use tensors but I was just wondering ( which I didn't state ) if the equation would be a rough estimate of the deformation on the sphere in the direction of x if x was taken to be on the radius of the sphere. Would it be a very rough estimate?
 
  • #4
If x was taken to be on the same axis as the radius that is. I'm not sure if that in itself is correct though. haha:smile:
 
  • #5
Well the problem would be a rough estimate if instead of a sphere we use a spinning disk, right?
 
  • #6
I will just use the tensor equations instead. Algebra is too limited lol:smile: Thank you very much for your time! You were very helpful!:smile:
 
  • #7
A spinning disk would be a more manageable problem to get a Theory of Elasticity solution for. In the spinning disk problem, you have a state of Plane Stress(which simplifies things), and, in addition, there is only one dependent variable, namely, the radial displacement u. The circumferential hoop strain is u/r, and the radial strain is du/dr. The principal directions of stress and strain are in the radial and hoop directions. So you have an ordinary differential equation to solve for the radial displacement, rather than partial differential equations for two displacements. In the disk problem, the boundary condition at the edge of the disk is zero radial stress.
 
  • #8
Thank you very much! I appreciate your help very much! Thank you for providing insight on this situation:smile:
 
  • #9
Here's an even simpler problem you can try, and it's statically determinate (so you don't have to use Hooke's law). Instead of a disk, consider a ring of circular cross section rotating about its axis. The elements of the ring are obviously accelerating radially, but what is causing the acceleration? There is no radial spring in this picture. In this case, the hoop stress is what is solely responsible for applying the radial acceleration. Do a force balance on a small curved section of the ring (free body), and use Newton's second law to determine the hoop tension.

Chet
 
  • #10
I never thought about it like that!:smile: Thank you very much! That actually simplified my problem to a great extent! If we decided to use a sphere, would we be able to use the bulk modulus to determine the force stretching the deforming the sphere? And then we can equate that to the equation to the force equation of centrifugal force?:smile:
 
  • #11
quantumfoam said:
I never thought about it like that!:smile: Thank you very much! That actually simplified my problem to a great extent! If we decided to use a sphere, would we be able to use the bulk modulus to determine the force stretching the deforming the sphere? And then we can equate that to the equation to the force equation of centrifugal force?:smile:

No, you couldn't do it that way. The disk problem and the sphere problem are not statically determinate problems. In both these cases, you would need to use the tensorial form of Hooke's law involving stresses and strains, and expressed in terms of the Young's modulus and the Poisson ratio. In the disk problem there would be one displacement to solve for, namely, the radial displacement. In the sphere problem, there would be two displacements to solve for, the radial displacement and the latitudinal displacement. The spherical problem would involve 2 partial differential equations involving the two displacements. I have trouble believing that neither of these problems has been solved in the open literature. I suggest that, if you want to avoid the work of solving the sphere problem, you search the literature (which you should be doing anyway).
 
  • #12
Thank you very much:smile:
 

1. What causes a spinning sphere to deform?

The deformation of a spinning sphere is caused by the centrifugal force, which is the outward force acting on an object due to its rotational motion. This force increases as the speed of the spinning sphere increases, causing the shape of the sphere to distort.

2. How does the deformation of a spinning sphere affect its trajectory?

The deformation of a spinning sphere can affect its trajectory by causing it to wobble or deviate from its intended path. This is because the uneven distribution of mass due to the deformation can create imbalances in the forces acting on the sphere.

3. Is there a limit to how fast a spinning sphere can deform?

Yes, there is a limit to how fast a spinning sphere can deform. This is known as the critical speed, and it is the speed at which the centrifugal force is equal to the elastic force of the material. Beyond this speed, the sphere will undergo permanent deformation.

4. Can the deformation of a spinning sphere be controlled?

Yes, the deformation of a spinning sphere can be controlled through various methods such as changing the material of the sphere, altering its shape, or adjusting the speed of rotation. These methods can help minimize or even eliminate the deformation of the sphere.

5. How does the size of a spinning sphere affect its deformation?

The size of a spinning sphere can affect its deformation in two ways. Firstly, a larger sphere will typically experience more deformation than a smaller sphere at the same speed due to its greater surface area. Secondly, the material properties of the sphere may also play a role in determining the amount of deformation it undergoes.

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