̂ + ⃗ ∙ ̂ , etc. This is viable because every one of these quantities, some after taking vector products, are in the same horizontal plane. Nevertheless, signs need to be taken good care of, for example, ‘ Ω⃗ × ’ will become ‘ × Ω ’, while ‘ 𝑅⃗ × ’ is ‘ × (− 𝑅) ’. After all these preparation work, we are left with the following set of complex equations ̈ = ̇ = − 𝑅 𝛺 = ̇ + 𝑅 By differentiating (3) we can substitute into (2), eliminate ̇ , and obtain an expression for , which could in turn be eliminated. Finally, we reached an ordinary differentiation equation for z.
Ω 𝑅 2 +
̈ − ̇ = 0 It is obvious that the solution will take the form = + Ω + 2 In the case of a solid ball, = 2 5
𝑅 2 . If we substitute in initial location 0
, and
initial velocity 0 are looking for.
, we can eventually complete the equation of motion that we
0
0
= ( 0
+
) −
where = 2 7
Ω . This is an amazing result! Substituting with Euler’s formula, = cos + sin , we can return to the vector space. The marble ball traces a new circle on the turntable! More surprisingly, it has a constant period, which is 3.5 times the period of the turntable, regardless of the mass or radius of the ball, or its position or velocity on the surface. Now switch on your record player and pick a marble, and you will discover almost exactly what is described above, provided the resistive forces play a negligible part. The result is so unexpected that barely anyone could make the correct prediction in the beginning. Although the phenomenon is counter-intuitive, mathematics does not feel it this way. Maths has its own strict disciplines for us to follow, avoiding any potential biases; while it is flexible enough to provide shortcuts to the solution. People often recognize maths as a slave for sciences, and yet in this very physical problem, it is maths who takes the dominance. Without mathematics, science could hardly turn anything descriptive into definitive.
Above all, there lies the sense of joy and relief when a convincing mathematical solution brings a problem to a satisfactory conclusion.
References Behavior of a ball on the surface of a rotating disc Am. J. Phys. 62, 151 (1994)
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