Re: For gyroscopes, what happens with forces applied @ all 3 axis?

Hi John,

Let me start off by apologizing for the delay in answering your question... I was called out of town and just got back in. So, on to the question at hand. In order to see how a generic gyroscope might react, let's look at the familiar example of a rotating bicycle wheel. In the picture below, imagine that you're holding the bicycle wheel, which is spinning clockwise, along it's axis of rotation and are able to exert a torque by moving your wrist up (in the z-direction), right (in the y-direction), etc. (apologies for the bad drawing... I was hoping to pilfer a drawing from some web site but a quick search for gyros turned up pitiful results and I was forced to fall back on my artistic skills, such as they are :). Thus, the apparatus is free to rotate around the origin, O (your wrist), and when you flex your wrist upwards, you apply a force F_z on the axis of the wheel.

Since the wheel is spinning clockwise in the y-z plane, its angular momentum vector is pointed along the x-axis. Now, if you apply a force in the F_z direction, this will create a torque along the y-axis (since torque is a result of vector multiplication). This torque will have an associated angular momentum also pointing along the y-axis. The original angular momentum (pointing along the x-axis) will combine with the added angular momentum (along the y-axis) to create a net angular momentum in the x-y plane. The axis of the spinning bicycle wheel thus moves along the x-y plane. So, in essence, a force in one direction (here, along the z-axis), shifts the gyroscope in a direction different by 90^o from the applied force (here, in the x-y plane). Thanks to conservation of angular momentum, when you try to move the gyro upwards, it pulls to the right. Similarly, if you applied a force in the -y direction, the gyro would pull up.

So what happens if you apply a force to each of the axes? We've already covered the scenarios for two of them, namely F_z and F_y. The only other place to apply a force is along the x-axis, F_x. But when you apply a force along the x-axis, the torque (defined as F x r) is actually 0. This is because a force applied in the x-direction is in the same direction as the axis of rotation (i.e., sin(0^o) = sin(180^o) = 0). And if there's no torque, there's no effect on the angular momentum. In other words, F_x is along the same axis as the original angular momentum (whereas F_y and F_z were perpendicular to it). Hence, if you try to push the spinning device forward or backward, it will simply react by moving forward or backward. Thus, the forces in all 3 directions are accounted for. Well, I hope that helps... if you'd like to clarify/correct anything, I'd be more than happy to discuss this further at rickys@sethi.org.

Regards,


Rick.