Why Do Gyroscopes *Start* Precessing?

[eck]

I was thinking about gyroscopes to me, and it became apparent that I'm really not comfortable with why a gyroscope attached to a pivot, with its axis parallel to the ground, doesn't swing about the pivot like a pendulum. I remembered learning something to the effect that it has to do with the precession of the gyroscope, but I couldn't quite figure it out, so I got out my mechanics texbook (by Kleppner and Kolenkow), and read up on what they had to saw about gyroscopes. They do the standard derivation of the equation for the angular velocity of precession, but the derivation depends on the fact that the gyroscope is no longer in the transient state, and the gyroscope is already precessing with uniform angular velocity. My question, is how does a gyroscope start precessing in the first place? It seems like the gyroscope will initially start to fall, and somehow this motion will evolve into the precession of the gyroscope. This could make finding the general equation for the motion of a gyroscope significantly more difficult than the special case when the gyroscope is already precessing! Can anyone offer insight as to what is going on?

 

[James R]

My question, is how does a gyroscope start precessing in the first place?

The gyroscope has an external torque on it around the pivot, due to its weight. The torque causes an angular acceleration, which results in precession.

 

[learningphysics]

I was thinking about gyroscopes to me, and it became apparent that I'm really not comfortable with why a gyroscope attached to a pivot, with its axis parallel to the ground, doesn't swing about the pivot like a pendulum. I remembered learning something to the effect that it has to do with the precession of the gyroscope, but I couldn't quite figure it out, so I got out my mechanics texbook (by Kleppner and Kolenkow), and read up on what they had to saw about gyroscopes. They do the standard derivation of the equation for the angular velocity of precession, but the derivation depends on the fact that the gyroscope is no longer in the transient state, and the gyroscope is already precessing with uniform angular velocity. My question, is how does a gyroscope start precessing in the first place? It seems like the gyroscope will initially start to fall, and somehow this motion will evolve into the precession of the gyroscope. This could make finding the general equation for the motion of a gyroscope significantly more difficult than the special case when the gyroscope is already precessing! Can anyone offer insight as to what is going on?

I had a look at the table of contents for that book... You should have more detailed explanations further on... "dynamics of a rigid body"... as well as "Euler's Equations".

I found this link slightly useful... unfortunately many of the pictures are missing:
http://www.ann.jussieu.fr/~bacaer/Prog/perspec/Intro/intro/node4.html

 

[HackaB]

My question, is how does a gyroscope start precessing in the first place? It seems like the gyroscope will initially start to fall, and somehow this motion will evolve into the precession of the gyroscope.

this is a legitimate and well posed question. Gurus, Mentors, Science Advisors: here is your chance to shine...

 

[tdunc]

Simply put, because of friction. There is a certain velocity that must be achieved or maintained to avoid degenerate precession. Friction may be a constant force that acts to slow down angular momentum which in turn has the effect of not being able to cancel out the effects of gravity which applies an torque perpendicular to the axis of rotation leading to precesssion. Note that amount of precession is directly related to velocity and that no matter what velocity, precession is always > 0; How noticable the precession is depends. So really your question is false because it is always has precession. (You could however argue there is an upper limit velocity where precession is near absolute zero.)

http://en.wikipedia.org/wiki/Gyroscope
http://en.wikipedia.org/wiki/Precession

 

[HackaB]

I honestly do not see how the OP could have made his question more clear, yet everyone is avoiding it completely (except possibly for learningphysics, who advised that he read his book).

Friction may be a constant force that acts to slow down angular momentum which in turn has the effect of not being able to cancel out the effects of gravity which applies an torque perpendicular to the axis of rotation leading to precesssion.

That's exactly what he's asking. How does the torque lead to precession. He obviously knows that it does. He's asking how. From the moment that the gyroscope starts to tip over, what makes it start going sideways? A force? Before you answer "because the torque is in the horizontal plane", remember that the same is true for a non-spinning gyroscope, which falls over.

 

[tdunc]

Its crystal clear to me what is going on, your making it way more difficult than it is. Read those links and think about it somemore. Why would torque Not lead to precession and the How is self intuitive. What forces are on the left side rotate to the right side and balance or cancel each other out and given the velocity is great enough to do this in a timely manner otherwise gravity will overcome the balance of forces and "tip the scales". The torque is in the verticle plane btw, gravity is a downward force.

 

[tdunc]

I may want to add this

Gravity is a downward force, by changing your momentum to go with the force of gravity you lose weight (eg. when you freefall you are considered weightless)

Imagine a large precession, the side whose angle of momentum is towards the ground feels a force of gravity slightly less than the opposite side whose momentum is towards the sky. So in this case there is not an equal force on both sides, and only when you rotate the device does it achieve an equalibrium and the forces are close to equal, then we get no precession. If it is not rotating fast enough these forces are not equalized fast enough to prevent it from falling to one side or the other.

What makes it start to noticably fall to one side or the other is knowing that it has a precession greater than zero at all times, this precession increases when velocity decreases.

 

[tdunc]

Everytime I reply on these forums I think of something new or better to say afterwards. So my final word on it ;)


If you were to "perfectly spin" with no initial precession a "perfect top" on a "perfect surface" and given gravity is exactly uniform and unfluxuating, Then and only then would precession = zero no matter the velocity of the top; since we have a perfect everything, the top when idle could stand upright on its own.

So right there are some additional Hows; uneven geometry density distribution in top structure, a flux in the force of gravity, uneven surface, any initial starting precession < this is the most common and obvious but any single one of these could cause the top to "start to fall" . The most important thing to realize is that once Any precession occurs, then future precession is allowed. So if we disturb our perfect experiment however so slightly or unnoticable, then we set up the requirements for future precession to occur.

 

[HackaB]

As Tyler Durden might say, "The first rule of Wikipedia: don't believe anything you read on Wikipedia."

Those two entries are particularly unsatisfying. But at least now I know where you got your "facts". Oddly enough, the Wiki entry on torque looks pretty good. The person who wrote the one on the gyroscope should read it, and possibly the one on cross products while he's at it.

Would anyone else like to take a stab at it? Surely someone has a good explanation.

 

[Claude Bile]

The gyroscope must start spinning somehow right? Once someone applies an external torque, the gyro begins to precess. To get a general derivation one needs to consider the application of this torque (specifically, the effect of treating the angular velocity of the gyro about the horizontal axis as a time dependant variable and not a constant).

Also, when I say external torque, I do not mean gravity, I mean someone actually spinning the thing by hand (or an equivalent case). The effect of gravity, as has already been pointed out is causing a spinning gyro to precess.

Claude.

 

[HackaB]

The gyroscope must start spinning somehow right? Once someone applies an external torque, the gyro begins to precess.

Okay, so you are talking about the torque required to start the gyroscope spinning here? In that case, if there is no gravity and I hold the frame of the gyroscope perfectly still while I spin the wheel up with a string, will it start to precess when I let it go (assuming I did not impart some motion to it by letting go)?

add: I'm not sure I understood you correctly. When you said "spinning", were you talking about the wheel spinning on its axis? That's how I interpreted it.

 

[James R]

The rotational equivalent of Newton's 2nd law of motion is:

[tex]\vec{\tau}_{net} = \frac{d\vec{L}}{dt}[/tex]

Rearranging, we get:

[tex]d\vec{L} = \vec{\tau}_{net} dt[/tex]

which means that if we look at the angular momentum vector at time t=0 and then again at time dt later, the new angular momentum will be the original plus dL. The vector dL added to the original L is in the same direction as the external torque.

In the case of a gyroscope, the gravitational force causes a torque in the horizontal plane. If the gyroscope is not spinning, then this torque results, after time dt, in an angular momentum in the horizontal plane, which causes the gyroscope to fall over.

If, on the other hand, the gyroscope is spinning, then the torque adds an amount dL in the horizontal plane to the existing angular momentum vector (also in the horizontal plane, but at right angles to the torque), which produces a new angular momentum vector in the horizontal plane. Since the angular momentum of the gyroscope must point along the axis of rotation, the gyroscope precesses as the direction of the angular momentum changes due to the gravitational torque.

Which is the long version of what I said above...

 

[Claude Bile]

Okay, so you are talking about the torque required to start the gyroscope spinning here? In that case, if there is no gravity and I hold the frame of the gyroscope perfectly still while I spin the wheel up with a string, will it start to precess when I let it go (assuming I did not impart some motion to it by letting go)?

add: I'm not sure I understood you correctly. When you said "spinning", were you talking about the wheel spinning on its axis? That's how I interpreted it.

In the absence of gravity, the gyro will not precess.

I interpreted the OP's question as 'how does one get from a non-precessing motion to a precessing one'. Perhaps eck could clarify.

Claude.

 

[HackaB]

I interpreted the OP's question as 'how does one get from a non-precessing motion to a precessing one'. Perhaps eck could clarify.

That's how I interpreted it too. Let's assume we're right. What did your answer have to do with his question? In other words, what does the initial torque required to start the wheel of the gyroscope spinning have to do with why it precesses when you put it in a gravitational field, with one end supported on a table?

 

[PhilG]

The rotational equivalent of Newton's 2nd law of motion is:

[tex]\vec{\tau}_{net} = \frac{d\vec{L}}{dt}[/tex]

Rearranging, we get:

[tex]d\vec{L} = \vec{\tau}_{net} dt[/tex]

which means that if we look at the angular momentum vector at time t=0 and then again at time dt later, the new angular momentum will be the original plus dL. The vector dL added to the original L is in the same direction as the external torque.

In the case of a gyroscope, the gravitational force causes a torque in the horizontal plane. If the gyroscope is not spinning, then this torque results, after time dt, in an angular momentum in the horizontal plane, which causes the gyroscope to fall over.

If, on the other hand, the gyroscope is spinning, then the torque adds an amount dL in the horizontal plane to the existing angular momentum vector (also in the horizontal plane, but at right angles to the torque), which produces a new angular momentum vector in the horizontal plane. Since the angular momentum of the gyroscope must point along the axis of rotation, the gyroscope precesses as the direction of the angular momentum changes due to the gravitational torque.

Which is the long version of what I said above...

Isn't this the explanation for why a gyroscope that has always been precessing must keep on precessing?

 

[James R]

Isn't this the explanation for why a gyroscope that has always been precessing must keep on precessing?

Well, yes, but it also accounts for why it starts to precess. Hold the gyroscope still, then let it go. It will start to precess for exactly the reason I gave. The angular momentum is not conserved, due to the external torque.

 

[da_willem]

Well, yes, but it also accounts for why it starts to precess. Hold the gyroscope still, then let it go. It will start to precess for exactly the reason I gave. The angular momentum is not conserved, due to the external torque.

But when the axis of a perfectly symmetric gyroscope is exactly aligned with the gravitiational field there is no torque:

[tex]\vec{\tau}=\vec{r} \times \vec{F} = 0 [/tex]

But when the axis is only slightly off, or it its mass distribution is not perfectly axissymmetric the gyroscope starts precessing. The question of the OP is quite like the question of why does a nonrotating gyroscope fall? Because it is in an instable equilibrium! When the axis of the spinning gyroscope is not perfectly perpendicular to the floor, note that this can also be caused by things like wind or the surface on which it spins, it starts precessing. At first with a very small radius, but then the gravitiational field can exert a torque which increases this radius.

 

[PhilG]

Well, yes, but it also accounts for why it starts to precess. Hold the gyroscope still, then let it go. It will start to precess for exactly the reason I gave. The angular momentum is not conserved, due to the external torque.

But the vertical component of angular momentum must be conserved, and in your explanation it is not. If the gyroscope starts precessing uniformly as soon as you let it go, there will be a increase (or decrease) in the vertical component of angular momentum associated with this motion. Where could that come from? Not gravity...

Of course, I can't offer any explanation better than yours

 

[PeteSF]

The original question mentioned a gyroscope with axis parallel to the ground - at 90° to the gravity field.

 

[James R]

But the vertical component of angular momentum must be conserved, and in your explanation it is not. If the gyroscope starts precessing uniformly as soon as you let it go, there will be a increase (or decrease) in the vertical component of angular momentum associated with this motion. Where could that come from? Not gravity...

That's almost right...

Actually, things are a little more complicated than I have explained.

The end of the axis of the gyroscope doesn't actually trace out a horizontal circle as the gyroscope precesses, but instead executes a series of small dips, in a general circle. This is hard to describe without diagrams.

If you're interested, try searching the web for "nutation". (In fact, I might do that myself and see what I can find.)

 

[Chronos]

What does a gyroscope precess relative to? If my coordinate system resides on the outer arm, the gyroscope appears to be stationary and all the usual laws of physics apply.

 

[PhilG]

That's almost right...
If you're interested, try searching the web for "nutation". (In fact, I might do that myself and see what I can find.)

Okay, will do.

Chronos: what are you getting at?

 

[reilly]

eck -- Your guess on how precession can start is right on. The basic physics of this precession can be found in almost any freshman physics book. The trick is to recognize that the downward pull of gravity, creates a tourque about any axis perpendicular to the vertical axis of the top. For example, get the top spinning, put on table, push the axis off vertical and let it go to precession. When tilted, the force of gravity and the top's major axis define a plane, and the tourque of gravity is then perpendicular to that plane as is the acceleration of precession.

The details are tricky and difficult, and are best handled with the Euler Equations. -- elliptic integrals and the like. These equations are needed to get the balance and flow of energy from rotational mode to rotational mode -- that happens during the up-and-down motion during precsssion. My references of choice are Goldstein's Mechanics, and Whittaker's A Treatise on the Analytical Dynamics of Particles and Rigid Bodies.
Regards,
Reilly Atkinson