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Given a coordinate system, where the $x$- and $y$-axes are on the horizontal plane and the $z$-axis is vertical.

Assume I launch a ball along the $y$-axis with backspin and sidespin. Immediately after being launched, the backspin is around the $x$-axis and sidespin is around the $z$-axis. As the ball flies through the air, the ball will curve right (or left) due to the sidespin. After flying and curving for a while, the ball will no longer be heading straight down along the $y$-axis as it was when launched. Will the axis of rotation change, for example, to match the one of the ball, or will it still be spinning around the $x$-axis?

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  • $\begingroup$ The ball is rotating around a single axis at any given point in time. It's not clear at all what you mean by it having backspin around one axis and simultaneously sidespin around a different axis. To be clear, the axis of rotation may or may not align with one of your chosen coordinate axes, but there is a single axis of rotation at any given time. $\endgroup$ – Brick Oct 26 '20 at 14:32
  • $\begingroup$ This also depends on the type of ball and how the axis of rotation is oriented relative to the irregularities on the surface of the ball. For example with a baseball, the spin on the ball matters but so does the orientation of the seams. A four-seam fastball and a two-seam fastball are thrown with essentially the same arm motion and same initial orientation of spin to the path of the ball, but they behave differently in flight because the pitcher holds the ball in a different orientation at release. A perfectly spherical ball w/ low friction, OTH, may not matter the spin at all, etc. $\endgroup$ – Brick Oct 26 '20 at 14:56
  • $\begingroup$ @Brick That's not necessarily a good way to look at it. Take a look at the gadget used to test astronauts, where they sit in a chair that is rotated around all three axes simultaneously. I am not sure but suspect you can't find a single axis about which points on a sphere revolve in circular motion. $\endgroup$ – Carl Witthoft Oct 26 '20 at 15:32
  • $\begingroup$ @CarlWitthoft I've been in that chair (multi-axis trainer) and it's a wild ride. But I'm not sure of your point. In the MAT you get a set of torques, and the torque around the axis of the inner wheel is wrt the rotating frame attached to the next outer wheel. So you can talk about spins along different axes b/c each axis is defined in a different rotating frame. Even on the MAT, you have an instantaneous angular rotation at any point in time in the frame of the room, although on MAT it is rapidly changing. $\endgroup$ – Brick Oct 26 '20 at 15:50
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    $\begingroup$ @Brick Yes, technically there is only one axis of rotation. What I'm trying to convey is that this spin can be broken into two components. The amount of spin around the X axis and the amount of spin around the Z axis. Regardless, my question is does the axis of rotation change as the ball curves or does the axis stay the same? $\endgroup$ – Chip Burwell Oct 26 '20 at 17:02
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Based on the comments, I interpret your question to be whether the axis of rotation changes at all as the ball curves rather than just some component of the axis of rotation changes, which in the original writing I found hard to understand.

The answer to that depends on a couple of things:

  1. What type of ball?
  2. How precise do you want to be about it?

For most rotating (mostly-spherical) balls in flight that curve, the primary force causing the curved trajectory is called the Magnus force. It's a result of an interaction between the surface of the ball and the air through which the ball is passing. For the ball, it will typically be through the center of the ball with a magnitude $F \propto \| \omega \times v \|$, where $\omega$ is the angular velocity vector and $v$ is the linear velocity of the (center of) the ball. Because this acts through the center of the ball, it will not generate a torque on the ball, and without a torque angular momentum is conserved, which means the axis of rotation does not change.

But if you need to be more precise, the fact is that real balls aren't exactly spherical - in fact most balls for sports deliberately introduce or accentuate some surface features to ensure that they are not spherical. For example, seams on a baseball are raised, golf balls have dimples, soccer balls are made from panels and so are not quite spherical and have seams. These deviations may (a) introduce perturbations to the form of the Magnus force or (b) create asymmetric drags on the ball as it rotates and flies. These differences need not effect different parts of the ball symmetrically because the ball itself is not symmetric, and so they might introduce a torque. At minimum we would expect a torque that slows the rotation due to friction, but the torque might have components that change the axis of rotation.

I infer that you meant a (mostly) spherical ball, but if you think of oblong balls (e.g. football or rubgy), the axis of rotation can change quite a bit in flight, and thinking of a more extreme case might help with the spherical case that I think interested you. Footballs may or may not have the nose drop noticeably during flight, a phenomena that depends on several factors including the speed of the ball and the rotational rate of the ball along the long axis at release. Footballs also sometimes precess in flight, where their axis of rotation is visibly changing during flight but in a (mostly) periodic fashion. Notably for your question, the precessing ball would happen because of a small amount of what you're calling backspin caused by the ball being thrown with its long axis misaligned to its direction of flight. So it's not just the spin on the ball but the orientation of the ball at release compared to its direction of flight.

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