Who among us hasn’t cast a spinning top onto the ground while twirling its axis, making it spin without falling over (at least during the first moments of its rotation). You have surely noticed that when the top is launched tilted and spinning, its axis slowly circles around a vertical axis relative to the ground, even as it keeps spinning on its own! And even more impressive, sometimes the top begins to oscillate vertically while it spins! What underlies these fascinating phenomena? And is there a link between our delightful top and the motion of planets? Join us and find out.

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Historical Background

The Hebrew name for the dreidel, “sevivon,” comes from the root ס.ב.ב (“to turn”), since we simply spin it. In English, a generic spinning toy is called a “spinning top,” whereas the top associated with Hanukkah is called a “dreidel,” a word of Yiddish origin meaning “to spin.” In Jewish tradition, the dreidel is linked to the era of Antiochus IV, when Jews studied Torah in secret; when discovered, they pretended merely to be playing with spinning toys. In the Middle Ages, the dreidel also served as a gambling device.

A Bit About Angular Momentum

After this brief historical survey, it is time to dive into the physics behind the dreidel, which is based on angular momentum. But first, what is momentum?

Momentum (or linear momentum) is defined [1] [2] as a property of a moving body that depends on its mass and velocity. Precisely, for a body of mass m moving with velocity v, its momentum p is p = mv. Velocity v is a vector, represented by an arrow whose length gives the speed and whose direction indicates the line of motion. Multiplying the mass by the velocity vector yields a new vector p, whose magnitude equals mass times speed and whose direction matches that of the motion.

By definition, an object with large mass but moderate speed (for instance, a truck) can have the same momentum as one with a small mass but high speed (such as a bullet). You can think of momentum as a measure of the “oomph” of motion.

So how does this definition apply to spinning objects? Up to now we have only described linear momentum. Linear momentum describes the motion of point-like bodies—objects whose motion can be represented by a single point moving in space, usually chosen as the center of mass. Many real objects move in far more complex ways; the simplest example is rotational motion. To describe such motion we define angular momentum [3]. For a point-sized mass, classical mechanics defines angular momentum as the cross product of the linear momentum p with the position vector r drawn from some reference point to the body: L = r × p.

Before going further, let us clarify the term “cross product.” In general, the cross product of two non-parallel vectors is a third vector whose magnitude depends on the magnitudes of the vectors and on the angle between them, while its direction is perpendicular to the plane containing the two original vectors. For example, if p points north and r points east, then L points upward, perpendicular to the horizontal plane of the compass rose.

Similar to linear momentum, angular momentum expresses the rotational “oomph” of a body. Consider a child sitting on a spinning carousel. At every instant the child has linear momentum determined by the combined mass (child plus carousel) and by the tangential velocity. Yet because of the continuous rotation, the direction of that velocity (and therefore of p) changes constantly. Likewise, the vector r from the carousel’s axis to the seat changes constantly. Hence, we need the angular momentum, whose direction (check the definition!) always points perpendicular to the plane of rotation, even though r and p are continually changing.

Angular momentum depends on two factors:

1. Angular velocity: the rate at which the angle swept by a rotating body changes relative to an arbitrarily chosen zero angle [4]. This quantity describes objects rotating about some axis, just as linear velocity describes straight-line motion.
2. Moment of inertia: the rotational analog of mass, but one that weights the mass distribution by its distance from the axis of rotation, and therefore represents the object's resistance to changes in angular velocity [5, 6]. Just as linear acceleration is produced by forces (gravity, electrical forces, etc.), angular acceleration is produced by “twisting forces,” i.e., forces that act at some distance from the axis and not along a radial line. Conveniently, the torque of a given force is defined as the cross product of the force with its lever arm—the vector from the axis of rotation to the point where the force is applied [7]. Sound familiar? That lever-arm vector is precisely the r in the classical definition L = r × p.

Whenever we study rotational motion we can therefore move from the classical definition (in terms of p and r) to an effective description (in terms of angular velocity and moment of inertia).

Armed with these definitions, we can finally understand how a dreidel moves. Let us perform a simple experiment: spin a dreidel on the floor so that its axis is not perfectly vertical [8], and ask what forces act on it. The weight of the dreidel acts on its center of mass, pulling it downward, while the normal force from the floor acts upward at the point of contact. These two forces cancel each other, so the dreidel neither rises nor falls while spinning. Despite this cancellation, gravity exerts an uncompensated torque that is solely responsible for the interesting effects we are about to examine.

Precession

In the first moments of motion the dreidel spins rapidly about its axis of symmetry, giving it angular momentum we call “spin.” Because gravity applies an uncompensated torque, this torque changes the spin direction constantly: the axis of symmetry slowly sweeps out a cone around the vertical, while the fast spin gradually slows. This slow circling is precession. If you are still not convinced, check out MIT's awesome video of a bicycle-wheel gyroscope [9].

Nutation

Now for another fascinating phenomenon. Take the precessing dreidel, spinning steadily about its symmetry axis, and tap its rim downward. At first the tilt angle increases, but the larger angle increases the torque and hence the precession rate. The faster precession then reduces the tilt angle, which in turn decreases the torque and slows the precession. This sequence repeats periodically, producing an additional oscillation of the dreidel’s axis about its mean angle. This motion is called nutation, illustrated (along with precession) in the attached video [10].

The Link to the Planetary Motion

Is there a connection between a small dreidel and Earth or other planets? Indeed there is. Planets such as Earth rotate about their axes (spin), making them giant tops. Unlike our toy, however, planets are not resting on a surface, and the simple constant gravitational force we assumed for the dreidel does not apply. Instead, planets feel gravitational forces from the Sun and all other planets. Combined with Earth’s slight equatorial bulge, these forces produce planetary precession.

Earth’s precession [11] has a period of about 26,000 years, so it does not affect our daily lives, but over long timescales it has intriguing consequences—one being that thousands of years from now Polaris, the current North Star (nearly aligned with Earth’s spin axis), will relinquish its title to another star.

In summary, this post highlights two captivating facts!
1. A seemingly simple object like a dreidel embodies deep, interesting physical phenomena that have many practical applications.
2. From human scales (a dreidel) to astronomical scales (the planets), the laws of motion are the same. It will be fascinating to discover how these laws might change when we probe the smallest scales—molecules, atoms, nuclei, and beyond… but that, perhaps, is a topic for another time.

We hope you enjoyed this post, and we wish you a happy Hanukkah!

English editing: Elee Shimshoni

References:

1. Momentum entry in Merriam-Webster Dictionary
2. Momentum entry in Reference.com Dictionary
3. Angular momentum definition, Wikipedia
4. Moment of inertia, The Free Dictionary
5. Inertia, Wikipedia
6. Angular momentum, Georgia State University HyperPhysics
7. Torque, Wikipedia
8. Precession of a Spinning Top
9. MIT Physics Demo — Bicycle Wheel Gyroscope, YouTube
10. Precession and Nutation, YouTube
11. Precession of the Earth, YouTube

Further Reading:

An excellent textbook on analytical mechanics:
Louis N. Hand & Janet D. Finch — Analytical Mechanics (Cambridge University Press, 1998).