how far are spinning tops influenced by the pi number ?

**Short answer:** Pretty far. Now for the long answer...

**Angular speed:** Our friend

*pi* enters top physics at many points, starting with the conversion of rotational frequency

*N* in RPM to angular speed

*w* in rad/s...

*w* =

*pi* *N* / 30 ~

*N* / 10

Hence, a top spinning at

*N* = 1,000 RPM has an angular speed of

*w* = 105 rad/s. The latter is the natural measure of rotational speed in physics and engineering and the only one routinely used in the relevant equations.

**Linear (circumferential) speed:** Every material point on or within a perfectly rigid top has the same angular speed

*w*, but its linear or circumferential speed

*u* grows with radius

*r* from the spin axis according to

*u* =

*r* *w*So a marking located

*r* = 0.100 m (100 mm) from the center of a top spinning at

*w* = 105 rad/s (1,000 RPM) streaks through a given line of sight at a whopping

*u* = 10.5 m/s (37.8 kph), while a marking located at half the radius goes by at half that speed. These speeds are plenty fast enough for optical illusions based on flicker fusion (aka persistence of vision).

Topmakers have been exploiting the power and versatility of the spinning top as an illusion-generating platform for centuries, and to wonderful effect. The reason is simple: A top packs a lot of linear speed across the visual field in a very small space. Pi enters the picture through

*w*.

**Centrifugal and Coriolis forces:** Every small mass

*m* making up a top experiences an outwardly radial centrifugal force

*F*_{cen} given by

*F*_{cen} =

*m* *w*^{2} *r*,

where

*r* is the radius from spin axis to

*m*, and

*w* is the top's angular speed. The dependence on the square of

*w* means that every top will turn to dangerous shrapnel at some speed. I'm careful to make my LEGO tops safe for speeds well above those attainable by hand. Designing for safety with a high-speed starter is a much bigger challenge.

Fighting centrifugal force may be a fact of life in topmaking, but you can still have fun with it by incorporating parts that move outward under centrifugal force in controlled ways. The LEGO blacklight centrifugal top below is a case in point.

https://www.youtube.com/watch?v=tG-szSIt84AWhen centrifugal force drives one of the freely swinging arms outward at speed

*v*, the arm also experiences a lesser sideways Coriolis force

*F*_{cor} perpendicular to

*F*_{cen}. The magnitude of the Coriolis force is given by

*F*_{cor} = 2

*m* *w* *v*,

where

*m* is now the mass of the arm. Centrifugal and Coriolis forces involve

*pi* through

*w*.

**Volume and mass:** The mass

*M* of a top of uniform density

*D* and volume

*V* is

*M* =

*D* *V*If the top is hollow,

*V* includes only the solid parts -- e.g., walls and tip in a hollow peg top. If the top has a shape of revolution with a relatively simple mathematical description, the volume generally has the form

*V* =

*pi* *B*,

where

*B* is a function of maximum radius

*R*, axial length

*L*, and perhaps other parameters (mostly top proportions), but not of

*pi*. For example, if the top is a uniform solid cylinder with stem and tip of negligible mass, then

*V* =

*pi* *R*^{2} *L* =

*pi* *B**B* =

*V* /

*pi* =

*R*^{2} *L**M* =

*D* *V* =

*pi* *D* *R*^{2} *L* =

*pi* *D* *B*Hence,

*pi* enters many top-related equations via

*M*. Contact processes produce braking torques that generally grow with

*M*. Tip friction is only the beginning here.

**Moments of inertia:** A real top's spin-down behavior is controlled mostly by its CM-contact distance

*H* (aka CM height), its axial moment of inertia

*I*_{3} about the spin axis, its transverse moment of inertia

*I*_{1} about the CM, its tip radius of curvature, and its aerodynamics. For basic top shapes, these moments of inertia can always be written in the form

*I*_{k} =

*M* *R*^{2} *G*_{k},

where

*k* points to the axis of interest,

*M* is the top's total mass,

*R* is its maximum radius from the spin axis, and

*G*_{k} is a dimensionless geometric factor depending only on top proportions and the axis of interest.

For the cylindrical top example above,

*G*_{3} = 1/2, and

*G*_{1} = (3 +

*a*) / 12, where

*a* =

*L* /

*R* is the cylinder's aspect ratio. For a thin circular disk with

*a* << 3,

*G*_{1} = 1/4. Since these geometric factors never involve

*pi*, the moments of inertia will involve

*pi* only if

*M* does through its volume

*V*.

**Precession and nutation rates:** The usual formula for precession rate

*p* (an angular speed) is

*p* =

*M* *g* *H* /

*I*_{3} *w*_{3},

where

*g* is the acceleration of gravity,

*H* is the CM-contact distance, and

*w*_{3} is the total angular speed about the spin axis. Our friend

*pi* enters implicitly in

*p* and

*w*_{3}. Any occurrences in

*M* and

*I*_{3} cancel.

On the other hand,

*pi* cancels completely out of the following formula for nutation rate

*n* in fast tops if you (i) consider that

*n* is also an angular speed and (ii) include all implicit occurrences of

*pi* on both sides.

*n* = (

*I*_{3} /

*I*_{1})

*w*_{3}**Critical angular speed for stable steady precession:** Writing this critical speed as

*w*_{3C}, we have

*w*_{3C}^{2} = 4 (

*I*_{1} +

*M* *H*^{2})

*M* *g* *H* cos(

*A*) /

*I*_{3}^{2},

where

*I*_{1} is still about the CM, and

*A* is the tilt angle. Though

*pi* completely cancels out of the right-hand side of this equation, it's still implicit in

*w*_{3C} on the left.

Note that critical speed depends most strongly on

*H* and not at all on

*M*, as mass cancels out of this equation completely. To see this, rewrite the moments on inertia as

*I*_{k} =

*M* *R*^{2} *G*_{k}as I did above and define the relative CM-contact distance

*h* =

*H* /

*R*,

where

*R* is still the top's maximum radius. Critical speed is then given by

*w*_{3C}^{2} = (4

*g* /

*R*) (

*G*_{1} +

*h*^{2})

*h* cos(

*A*) /

*G*_{3}^{2}At this point, nothing depends on mass. And the only parameter depending on the top's absolute size is

*R*, as the geometric factors

*G*_{1} and

*G*_{3} depend only on top proportions.

**Aerodynamic braking torque:** The von Karman formula for the aerodynamic braking torque exerted on the faces of a spinning disk also includes

*pi* -- both explicitly and implicitly via

*w*. The same is likely to be true for other top shapes. I'll spare you the math.

**Bottom line:** Tops are crawling with

*pi*.