Don't quite understand precession

[Whitebread1]

I don't quite understand WHY a gyroscope precesses the way it does.
I'll set up an example in order to ask the questions I need.

Lets say you have a spinning bicycle wheel suspended at one end of its hub by a string. Due to the angular momentum of the spinning wheel, the wheel will precess about an axis parallel to the string.

I understand that the wheel has an angular momentum vector parallel to the axis the wheel spins about (horrizontal). And I understand that the weight of the wheel produces a torque vector that is orthogonal to the axis of rotation of the wheel. What I don't understand is how the torque vector pointing in this direction causes precession about the string. I don't see how a torque can cause rotation about an axis oriented 90 degrees away from the torque vector.

Once the wheel starts to precess about the string, the momentum vector changes position and rotates about the string along with the axis of rotation of the wheel. How does this motion not violate the principal of conservation of angular momentum when the movement of the rotation axis downward does?

 

[OlderDan]

Take a look at this thread

https://www.physicsforums.com/showthread.php?t=143246&highlight=momentum

If that does not clear it up for you, then at least you will have seen the diagram and we can take it from there. Do you see the connection between the diagram and the example you have cited?

 

[Whitebread1]

I understand that diagram. The weight is pointing down, the radius is parallel to the axis of rotation, so torque is orthogonal, as the diagram shows. What I don't understand is where the force that causes the precession comes from. It is my understand that when something rotates about an asix, the force that causes the rotation is in the plane of rotation. In the example of the bicycle wheel, the only external force I see is gravity.
Another way to put it:
The torque and dL vectors are alwasy parallel. The fact that the bike wheel is spinning about the string implies there is an angular momentum and a torque vector parallel to the string. Inducing a torque in this direction would require a force to act on the end of the bike wheel hub in the horrizontal direction. Where does this force come from?

 

[OlderDan]

I understand that diagram. The weight is pointing down, the radius is parallel to the axis of rotation, so torque is orthogonal, as the diagram shows. What I don't understand is where the force that causes the precession comes from. It is my understand that when something rotates about an asix, the force that causes the rotation is in the plane of rotation. In the example of the bicycle wheel, the only external force I see is gravity.
Another way to put it:
The torque and dL vectors are alwasy parallel. The fact that the bike wheel is spinning about the string implies there is an angular momentum and a torque vector parallel to the string. Inducing a torque in this direction would require a force to act on the end of the bike wheel hub in the horrizontal direction. Where does this force come from?

For a wheel with a horizontal axis suspended by a string at one end of the axis there are two forces acting on the system. One is gravity which can be taken as acting at the CM of the wheel. The other is the tension in the string. The diagram in that other thread is drawn from the perspective of looking down from above with the string attached at point O. If you calculate the torque about point O, the string force contributes nothing because r = 0. The weight of the object is acting at a distance D (2.0cm in the diagram) from O, so the torque is WD = MgD, where W is the weight and M is the mass of the wheel. The weight vector W is vertically downward and the vector D from O to the CM is horizontal. Torque is the vector cross product D x W which is perpendicular to both D and W (horizontal and perpendicular to the axis) with the direction given by the right-hand rule. The torque vector in the diagram correctly represents this torque.

The torque could also be calulated about the CM, in which case the weight contributes no torgue but the tension in the sting is applied a distance D from the CM. The vector from the CM to the string is -D and the force vector is upward and equal to the weight, -W. So the torque would be -D x -W = D x W the same as when calculated about point O. You could calculate the torque about any point and get the same result. It is a horizontal vector perpendicular to the axis of the wheel.

 

[Whitebread1]

I understand what you have written.

The precession of the wheel about the string means the system has an angular moment component parallel to the string. This means that a torque must have existed parallel to the string to start the precession in the first place. I am wondering where the torque for this precession came from.
I don't see how a change in angular momentum in the horrizontal plane can cause a change in angular momentum parallel to the string.

 

[cesiumfrog]

In first year I was given the explanation that begins by considering a small element of the gyroscope's circumference, and accounting for every force that applies to it..

 

[OlderDan]

I understand what you have written.

The precession of the wheel about the string means the system has an angular moment component parallel to the string. This means that a torque must have existed parallel to the string to start the precession in the first place. I am wondering where the torque for this precession came from.
I don't see how a change in angular momentum in the horrizontal plane can cause a change in angular momentum parallel to the string.

On the contrary, there has been no consideration of angular momentum parallel to the string. The angular momentum is horizontal. The torque is horizontal and perpendicular to the angular momentum at all times. The angular momentum changes direction but remains in the horizontal plane. The only things acting perpendicular to the horizontal plane are the forces that lead to the torque vectors that lie in the plane. It is the torque in the plane that accounts for the precession.

However, I think I see what is bothering you. As the wheel precesses there is a small horizontal velocity component of the wheel's mass and a slow rotation of the system about the vertical axis. This has been ignored in the previous discussion. To fully understand this you need to look deeper into the complexities of rigid body motion. In more advanced treatments of the subject, angular momentum is a tensor (matrix) that relates angular velocity to angular momentum. The two are not necessarily parallel. An angular velocity vector along an axis in one direction can lead to an angular momentum vector with components in all three directions. The full treatment is a complex subject, but you do not need all that complexity to account for the precession of angular momentum in a plane.

 

[daniel_i_l]

you can understand why a force pulling down (gravity) on the axis of the wheel pushes the wheel forward by looking at one little part of the wheel. let's look at the point that's currently at the bottom. gravity will cause the wheel to tilt around the z (in-out) axis, during the tilt the part that was on the bottom spins up to the top, but instead of going in a straight path (only spinning on the y axis) it goes in a curved path (it also spins around the z axis), this curved path implies a force - like the centrifugal force) pushing it backwards, but because there isn't a force in that direction it moves forward. so basically, the motion of the particles of the wheel in a curved path due to the tilt of the axis of rotation causes the precession.

 

[Whitebread1]

On the contrary, there has been no consideration of angular momentum parallel to the string. The angular momentum is horizontal. The torque is horizontal and perpendicular to the angular momentum at all times. The angular momentum changes direction but remains in the horizontal plane. The only things acting perpendicular to the horizontal plane are the forces that lead to the torque vectors that lie in the plane. It is the torque in the plane that accounts for the precession.

However, I think I see what is bothering you. As the wheel precesses there is a small horizontal velocity component of the wheel's mass and a slow rotation of the system about the vertical axis. This has been ignored in the previous discussion. To fully understand this you need to look deeper into the complexities of rigid body motion. In more advanced treatments of the subject, angular momentum is a tensor (matrix) that relates angular velocity to angular momentum. The two are not necessarily parallel. An angular velocity vector along an axis in one direction can lead to an angular momentum vector with components in all three directions. The full treatment is a complex subject, but you do not need all that complexity to account for the precession of angular momentum in a plane.

Yes, I figured the physics at work was far more complicated than my texts led me to believe. I think I will just leave it be until I study more complicated mechanics.
Thanks.

 

[Whitebread1]

you can understand why a force pulling down (gravity) on the axis of the wheel pushes the wheel forward by looking at one little part of the wheel. let's look at the point that's currently at the bottom. gravity will cause the wheel to tilt around the z (in-out) axis, during the tilt the part that was on the bottom spins up to the top, but instead of going in a straight path (only spinning on the y axis) it goes in a curved path (it also spins around the z axis), this curved path implies a force - like the centrifugal force) pushing it backwards, but because there isn't a force in that direction it moves forward. so basically, the motion of the particles of the wheel in a curved path due to the tilt of the axis of rotation causes the precession.

A write up on precession I found on the internet mentioned this in some way, but was not clear. Thanks.