Can an object spinning near the speed of light be accelerated significantly in any direction? This question made me curious. It refers to a ball which is rotating so that its equator is approaching the speed of light, then talks about what it would look like if the observer moved. What I'm curious about is if the ball itself could move much at all. If the edge is already moving very near the speed of light, would its rotation have to slow down for the whole ball to accelerate significantly in any direction?
For clarification, the way I'm thinking of this is that you'd have a ball (made of unobtanium - this questions isn't about whether or not it's possible) spinning so that the top speed of any point on it is very close to $c$ and otherwise not moving. Then you give it a strong shove (parallel to a line from its center to the "equator" of spin, for simplicity) while it continues to spin. Then, instead of all points along its equator moving at an equal speed, the side spinning towards the direction of travel would be changing position faster than the side spinning away from that direction. Assuming your shove was strong enough, that could push the forward-spinning edge across the speed of light.
Of course that can't happen, so the question is what would happen? Would the spin of the ball need to slow down to compensate, so that no part of it travels faster than $c$? Would it resist being pushed at all, the same way a gyro resists when you try to move it? Something else entirely? Am I even envisioning the physics of it correctly? If not, please give an explanation of where my reasoning is wrong.
 A: Revolutions per minute as measured by any clock attached to any part of the ball will be unchanged.
A clock on the equator will be experiencing varying time dilation after the push, in the frame where the ball has linear motion. A clock on a pole will experience just the most basic time dilation in the frame where the ball moves linearly.
Linear motion must not cause any accumulation of time difference between a clock on the equator and a clock on a pole. Generally a spinning ball must not be an detector of absolute linear motion.
If a clock on the equator measures how much time it spends on the left side of the ball and on the right side of the ball, the result must be that the times are equal.
If an observer in the frame where the ball has linear motion measures how much time a clock on the equator spends on the left side of the ball and on the right side of the ball, the times are not the same. The ratio of times spend on the left side and the right side must be the same as the ratio of time dilations on the left side and the right side. 
(Please somebody edit this ;) or tell me what to say instead of "left side" and "right side")
A short answer: It is possible to use a spinning ball as a clock. A spinning-ball-clock time dilates like clocks do. 
A: 
Of course that can't happen, so the question is what would happen? Would the spin of the ball need to slow down to compensate, so that no part of it travels faster than c? Would it resist being pushed at all, the same way a gyro resists when you try to move it? Something else entirely? Am I even envisioning the physics of it correctly? If not, please give an explanation of where my reasoning is wrong.

In this unnatural setting (where classical electrodynamics exists and no cohesive forces with electromagnetic exchanges hold the ball together,) what will be happening   as the equator of the  ball approaches c and the central parts around the axis, the relativistic mass will be growing the further out from the axis .
Special relativity states for masses moving with a high velocity close to the velocity of light that the inertial mass changes , it is called relativistic mass, given by :

a body at rest has the rest mass

with the ratio

gamma goes to infinity as the velocity approaches c, and in the scenario above that is what will be happening to the density at the rim:
If the ball started with density d grams per centimeter it will become a function, d(r) where r is the distance from the axis. It will be the rotation of a non uniform ball . If a tangential force is applied to increase the outside equator motion, the mass there will grow accordingly to the impulse and a wobble will enter the uniform rotation. 
In special relativity the extra energy entering a system when approaching the velocity c turns to mass , due to E=mc^2. 
I will note that this is one of the few cases where the concept of relativistic mass is useful, i.e when one applies Newtonian physics where the second law holds with the relativistic mass. 
A: 
Would it resist being pushed at all, the same way a gyro resists when you try to move it?

Let's say the ball has mass 1 kg when not spinning, and mass 10 kg when spinning.
When that spinning ball is pushed around, it will behave almost like a non-spinning ball with mass 10 kg.
When the spinning ball is pushed around at non-relativistic speeds, one can use Newton's laws to find the approximate trajectory of the ball. $ F=ma $ ,  where $ m=10 kg $ in our example case.
When the spinning ball is pushed around at relativistic speeds, one can use relativistic mechanics to find the approximate trajectory of the ball. $ F=ma* \gamma ^3 $ ,  where $ m=10 kg $ in our example case.
There is just a small subtlety: A ball that is spinning and moving linearly does not necessarily behave like it was a more massive non-spinning homogeneous ball, instead it may behave like it was a more massive non-spinning ball whose one side is more massive than the other side. 
