Understanding Precession

The central phenomenon in nuclear magnetic resonance is the precession of nuclear spins in a magnetic field. Precession is the same phenomenon that delights us in the curious behavior of gyroscopes and of tops in general. When you apply a torque aimed at changing the direction of the axis, the top responds by moving the axis in a different direction.

Here is a top. We have spun it clockwise, as shown by the curved arrow, and at an angle so that gravity is trying to twist the axis clockwise (arrow). We know from experience that instead of falling toward the table, the axis of the top will precess about a cone moving the handle around the red horizontal circle centered above the point of contact with the table.

Which direction will the handle move about this circle, clockwise or counterclockwise, i.e. in the figure will the handle move toward us or away?

Once you understand the reason for this curious response - moving one way when pushed another - you can easily predict which direction the axis will move without resorting to memorized formulas involving vectors and "right-hand" rules, which are too often the refuge of the ignorant. [Feynman says, "many simple things can be deduced mathematically more rapidly than they can be really understoood in a fundamental or simple sense...the precession of a top looks like some kind of a miracle involving right angles and circles, and twists and right-hand screws. What we should try to do is to understand it in a more physical way." (Lectures on Physics, Vol. I, p. 20.6)]

Stationary Wheel

Consider a stationary bicycle wheel on a blue axle suspended by a string on the left. Gravity pushing down (red arrow) and is converted by the force of the string into a torque, transmitted through the spokes, to push the top of the wheel to the right and the bottom of the wheel to the left (violet arrows).

Left to its own devices the wheel will flop down in the direction of the torque. The top of the wheel moves to the right and the bottom to the left, as expected.

There is no weirdness if the wheel is not spinning.


Spinning Wheel

Now consider a wheel that is spinning toward the viewer at the top (curved red arrow) and ask the question "Which part of the rim is moving fastest to the right?"

Remember that every portion of the rim above the height of the axle is feeling a force to the right, and every portion below the axle is feeling a force to the left.

Remember also that force is not velocity. It is proportional to acceleration - that is to the rate of change of velocity. Thus the longer time a piece of the rim has felt a force to the right, the faster it will be moving to the right.

As shown on the left, the piece of the rim nearest the viewer will have the largest rightward velocity (green arrow), because it has been accelerated to the right ever since it rose above the axle, half a revolution earlier.

By the same token, the piece of the rim furthest from the viewer will have the largest leftward velocity (orange arrow).

As the wheel's spin raises this furthest bit of the rim toward the top from the furthest back position, it is constantly experiencing a force to the right, which slows its motion to the left until, at the top, it is moving neither right nor left.

From the top to the front, the rim continues accelerating to the right, reaching its highest velocity to the right when it is closest to the viewer. Then it falls below the axle, and the force to the left slows motion to the right until, at the bottom, there is no right-left motion. Thus the half of the rim nearest the viewer is moving right and the far half is moving left.

The relative velocities in the figure on the right show a remarkable thing : the actual motion of the wheel is rotation about the vertical axis, not about the axis perpendicular to the page, about which gravity and the string apply a torque.

This simple idea explains what is most curious about steady precession. Describing exactly how the axis begins to move when the torque is first applied is a little more complicated. The initial jerky motion, observed most easily when the spin is slow, is called "nutation" and begins with a bit of falling down in the direction of the applied torque. [For an analysis of nutation, see Feynman Lectures on Physics, Vol. I, pp. 20.6-7.]


1) Which way should the handle of the top in the first figure move - toward the viewer or away?

2) Should increasing the torque by adding weight to the axle of the bicycle wheel increase or decrease the velocity of precession?
[This relates to variation in the nuclear precession frequency with magnetic field strength.]

3) Should increasing the rate at which the bicycle wheel spins increase or decrease the velocity of precession?

4) The figures above show the axle at 90° to the vertical string. How should setting the axle at other angles influence the rate of precession?
[This is a tougher problem because the angle influences two parameters: (1) the torque supplied by string and gravity,
and (2) the amount of redirection of the axle that is required to constitute a full cycle of precession.
You may want just to consider the direction of each influence.]

(Click here for Answers)


This is one of many cases which are confusing because of the "phase lag" between a driving force and its resulting integrated effect.

The graph plots the amount of force (right or left) and the velocity (right or left) as a function of position on the rim of the wheel in the diagrams above.

Because the "velocity wave" is 90° behind the "force wave" that generates it, the precession axis is 90° away from the torque axis.

Closely analogous phase lags between a driving force and its integrated effect are important in all sorts of phenomena, especially in electronics and optics.

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copyright 2003 J. M. McBride