angular momentumgyroscopesprecessionrotational-dynamics
What causes precession in a spinning object?
What causes nutation in a spinning object?
What causes a top, gyroscope, and the earth to wobble?
Just because it's a simple question, I'm not expecting a simple answer, but please do summarize whatever you say in laymen terms, thanks.
The body is rotating around two axis $x$ and $z$ So,the angular momentum along those direction get add up to $L_{resultant}$ which causes the rod to slightly dis-align from the $x$-axis as seen from the top along $y$-axis
In detail what happens is when we rotate the rod along $x$-axis and then apply a torque $\tau_z$ for small amount of time in $z$ direction which builds up the angular momentum in $z$ direction.By intuition we tend to think that the rod will continue spinning in $x$ axis and in the same time will rotate about $z$ axis.But that's not true because if rod rotates in $z$ direction then $L_x$ will change (in direction not in magnitude) without the application of any torque which violates the law of conservation of angular momentum.So instead of this ,the rod just tilts in the direction of resultant angular momentum(shown in diagram) and will remain until some other torque is applied. And moreover it will not wobble and will not have any angular velocity along $y$-axis since there is no torque in that direction.
See this
may be you will get better understanding.
On the other hand if you apply a constant torque along $z$-axis then $L_x$(lets call it $L_s$ where $s$ stands for spin) will rotate towards $z$-direction but now also it will not wobble.But if you somehow make sure to make $\tau_z$ always perpendicular to $L_s$(which is exactly the same case as gyroscope) then $L_s$ will try to chase $\tau_z$ hence rod will now have angular velocity $\omega _{pr}$($pr$ stands for precession) $$\omega_{pr}=\frac{\tau}{L_s}$$
And as your question, is it possible for a body to spin in $x$-axis and tumble in $z$-axis with no rotation in $y$-axis?
My answer is when you apply torque in $z$-direction it doesn't mean it will tumble (in the video, the wheel doesn't tumbled inspite of having a torque =$mgr$ ) instead it will have a angular velocity along $y$-axis $\omega_{pr}$.
Not easy to explain intuitively, but I'll give it a go. Let's use the following 3d model of the system at hand for reference (the Mathematica code used to make this model can be found in this gist):
Let in the above $\hat z$ be the direction from the base of the structure to the red sphere, and $\hat x$ the unit vector going from the red sphere towards the gyroscope.
The red arrows represent the velocity of various parts of the rotating gyroscope. The green arrows represent the effective action of gravity on the center of mass of the rod, and the corresponding counterforce provided the hinge.
The combined action of the two green arrows generates a torque on the rod + gyroscope system, trying to push it down. But the only way for this to happen, it for the gyroscope to rotate in the $\hat x \hat z$ plane:
The cyan arrows show the corresponding forces that this rotation induces on the upper and lower points of the gyroscope. Now, remember that the cyan arrows represent forces, while the red ones velocities. The cyan arrows will induce an acceleration on the the various points of the gyroscope. In particular, they will induce the red arrows to change direction. In the model this is shown for the upper and lower points, with the orange arrows represnting the modified velocity vectors in those points.
As you can see from there, the new velocity vectors correspond to the ones you have when the gyroscope changes its direction following the precession motion. Basically, the action of the cyan forces on the red velocities is what causes the precession motion.
Now for the nutation, we have to note that the cyan arrows above are not actually quite correctly drawn. The motion of the gyroscope falling dawn more correctly corresponds to the cyan arrows being slightly tilted towards the ground. This is of course to be expected: after all, if the gyroscope wasn't rotating, it would just fall down. This is why even though there is precession, the gyroscope does go down a bit.
This acceleration towards the ground, because of the same mechanism explained above, also induces a faster precession. But an accelerated precession now causes a counter-falling reaction, for reasons similar to the ones causing the precession itself.
An accelerating precession motion means that there are effectively force vectors pushing the gyroscope to rotate in the $\hat x \hat y$ plane. Again, this corresponds to force pushing different points of the gyroscope in different directions. In the following, the cyan arrow show the forces acting on two points:
Again, these cyan forces will induce a change in the direction of the corresponding red velocities, and the orange arrows show what the velocity will be soon after. As you can see, the new velocities are those corresponding to the gyroscope direction going upwards, thus generating the "counter-falling" phenomenon that is in this case the nutation.
Best Answer
There is an explanation of precession entirely in laymen terms in an earlier thread (answer by me, Cleonis, on december 26 of 2012) That earlier thread is What determines the direction of precession of a gyroscope?
Necessary and sufficient for an object to be in a state of gyroscopic precession: the object is spinning, and a torque is being exerted. In my explanation I show how that happens.
Nutation can be thought of as a form of overshooting. If you release a spinning top roughly then for sure there will be a nutation superimposed upon the precession. You can avoid nutation by releasing very gingerly. (In effect the careful release dampens extra motions.)
Generally the nutation motion of a spinning object will dampen out fairly quickly, the kinetic energy of the nutation motion dissipates relatively fast.
The nature of nutation:
I mentioned 'overshooting'.
Let's say you have a gyroscope set up with it's spin axis horizontal, so that the torque from gravity is maximal. Then you release the spinning top to just drop down.
- The center of mass of the spinning top then moves down. That downward motion induces precessing motion (see the linked to explanation)
- the precessing motion is so fast that the spinning top climbs again.
- The climbing motion reduces the precession rate.
- because of the reduced precession rate the center of mass drops down again.
So, the nutation is a cyclic energy conversion process:
- there is a stage of giving in to the torque that is being exerted.
- during that stage the torque is doing work and the kinetic energy of the spinning object increases, which goes to the precession rate.
- When the spinning object is climbing again its kinetic energy decreases again as it gains potential energy.
When nutation has dampened out then the rate of precession is precisely enough to keep gravity from pulling the center of mass of the spinning top lower.
Gyroscopic precession and nutation are intimately related. Nutation is inherently a perturbation of a state of precessing motion. That is, there is no such thing as nutation without gyroscopic precession.
Earth wobble
The Earth's spinning motion is very complicated. There is a torque from the Sun and a torque from the Moon, and the Moon's orbital plane is at an angle with the Earth orbit around the Sun. Probably those influences play a role in the Earth wobble.
General remark: It's obvious that the angular momentum of the spinning top is not conserved. You get gyroscopic precession when a torque is exerted. That torque causes change of angular momentum.
Still, stable gyroscopic precession is cyclic, so something is conserved there. The direction of the angular momentum is changing all the time; the thing that remains the same is the magnitude of the angular momentum.