Why is the change in angle of precession equal to dL/L?

In summary, the conversation discusses the precession of a rapidly rotating gyroscope attached to a horizontal perch. It introduces variables L for angular momentum, dL for a small change in momentum, and dA for the corresponding change in angle. According to references, dA is equal to dL divided by L. The speaker questions this and suggests that the change in sine should be used instead. However, the other person explains that when using radial coordinates, the change in angle can be approximated by dL over L.
  • #1
trobinson41
2
0
If a rapidly rotating gyroscope attached to a perch is released in a horizontal position -- that is, so that the axis of the gyroscope is horizontal -- the gyroscope will precess around it's perch. Let L represent its angular momentum. Let dL represent a small change in that momentum. Let dA represent the corresponding change in the angle of L. According to all the references I've found, dA = dL/L. I don't understand this. Shouldn't dL/L be the change in the sine of the angle, not the angle? See attached diagram. Thanks.
 

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  • #2
It looks like dL and dA are both measured in radial coordinates.
dL = L * dA will give you the appropriate measure to arrive at the point L(t + dt) in the illustration. If you were to use the sine, you would be headed to a point more than |L| away from the center.
 
  • #3
Your mention of radial coordinates jogged my memory. The angle in radians is the arc length / radius. The radius in this case is L. Since the arc length approaches dL as the length of dL decreases, dL/L approaches dA. I think that's the reasoning that's being used. Thanks for your help.
 

Related to Why is the change in angle of precession equal to dL/L?

1. Why is the change in angle of precession equal to dL/L?

The change in angle of precession is equal to dL/L, where dL is the change in angular momentum and L is the total angular momentum, because of the conservation of angular momentum. This law states that the total angular momentum of a closed system remains constant, meaning that any changes in one component must be balanced by changes in another component.

2. How does conservation of angular momentum relate to the change in angle of precession?

The conservation of angular momentum is directly related to the change in angle of precession. This is because the precession of a spinning object is caused by a change in its angular momentum, and according to the conservation law, this change must be equal to the change in the total angular momentum of the system.

3. Can you explain why the change in angular momentum causes precession?

The change in angular momentum causes precession because of the gyroscopic effect. When a spinning object experiences a change in its angular momentum, it also experiences a change in its axis of rotation. This results in a precession motion, where the axis of rotation of the object moves in a circular motion around a fixed point.

4. Is the relationship between change in angle of precession and dL/L always true?

Yes, the relationship between the change in angle of precession and dL/L is always true as long as there are no external torques acting on the system. This is because the conservation of angular momentum is a fundamental law of physics that applies to all closed systems.

5. How does the mass and velocity of an object affect the change in angle of precession?

The mass and velocity of an object do not directly affect the change in angle of precession. However, they can indirectly affect it by changing the object's angular momentum. Objects with larger masses and higher velocities will have a greater angular momentum and therefore may experience a larger change in angle of precession under the same external torque.

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