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A spinning top looks like a toy, but writing down exactly how a heavy body turns about a fixed point is one of the hard problems of classical mechanics. Before 1888 it had been solved in only two special cases. Sofia Kovalevskaya found a third, and she did it after being refused entry to university lectures because she was a woman. This post tells her story and explains her result using the moment of inertia you study in Class 11.
Who was Sofia Kovalevskaya?
Kovalevskaya was born in Moscow on 15 January 1850 and died in Stockholm on 10 February 1891, according to her MacTutor biography. In 1869 she went to Heidelberg to study mathematics and science, only to find that women could not matriculate. She persuaded the university to let her attend lectures unofficially, with each lecturer's permission.
In 1871 she moved to Berlin to study with Karl Weierstrass. The university senate refused to let her attend lectures, so Weierstrass taught her privately. By spring 1874 she had written three papers, on partial differential equations, Abelian integrals and Saturn's rings, and Weierstrass judged each worthy of a doctorate. Göttingen University awarded her the degree in 1874, summa cum laude and in absentia.
A doctorate did not bring a job. MacTutor describes years in which she could not find an academic position. Gösta Mittag-Leffler finally secured her a post at Stockholm: she began lecturing there in early 1884, at first without pay, and in June 1884 was appointed to a five-year professorship. In June 1889 she became, in MacTutor's words, the first woman since Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university.
What problem did she solve?
The French Academy of Sciences set its Prix Bordin competition, announced in 1886, on the motion of a rigid body. Kovalevskaya's summary of her work, published in Acta Mathematica in 1889, opens with the state of the problem. An English translation posted on arXiv in 2026 makes it readable today.
She writes the body's principal moments of inertia about the fixed point as A, B and C, and the position of its centre of gravity as x₀, y₀, z₀. Until then the equations had been integrated in only two cases:
Euler's (or Poisson's) case: the centre of gravity is at the fixed point, so gravity exerts no torque.
Lagrange's case: A = B, with the centre of gravity on the symmetry axis. This is the ordinary spinning top.
She looked for cases where the solutions behave as smoothly as these two, and found exactly one more: A = B = 2C, with the centre of gravity in the plane perpendicular to the special axis (z₀ = 0). A footnote in the paper records that the Academy, at its session of 24 December 1888, awarded the work the Prix Bordin, raised from 3,000 to 5,000 francs. MacTutor adds that fifteen papers had been entered.
Worked example: reading moments of inertia like Kovalevskaya
Her condition is about ratios of moments of inertia, so it helps to compute some. Use the values in Table 6.1 of the NCERT Class 11 Physics chapter on rotational motion. Take a sample disc of mass 0.1 kg and radius 0.05 m.
About its axis (perpendicular to the disc, through the centre): I = MR²/2 = 0.5 × 0.1 × 0.0025 = 1.25 × 10⁻⁴ kg m².
About a diameter: I = MR²/4 = 0.25 × 0.1 × 0.0025 = 6.25 × 10⁻⁵ kg m².
Ratio. The two diameter moments are equal, and the axial moment is twice each of them. In Kovalevskaya's letters, with C for the special axis, that is A = B = C/2.
Compare. Her case needs the opposite ratio, A = B = 2C: the special axis must have the smallest moment, half of the other two. A flat disc cannot be a Kovalevskaya top.
Angular momentum. If the disc spins about its axis at 30 rad/s, L = Iω = 1.25 × 10⁻⁴ × 30 = 3.75 × 10⁻³ kg m²/s.
The lesson for exams is the same one she used: the moment of inertia depends on the axis, and you must state which axis before you write a number.
Common mistakes students make with rotating bodies
Quoting I without an axis. "MR²/2" is the disc about its axis, not about a diameter.
Assuming symmetry means simple. A body can have two equal moments and still be hard to analyse if its centre of gravity is placed awkwardly, which is exactly the gap Kovalevskaya filled.
Mixing torque about different points. Take torques and moments of inertia about the same point or axis.
The JEE (Main) 2026 syllabus lists moment of inertia, radius of gyration and angular momentum in its rotational motion unit. Our guide to moment of inertia and the axis theorems covers the problem types.
Spinning motion is easier to understand when you can watch it. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
What can a student learn from how she worked?
Kovalevskaya did not try to solve the general problem by force. She asked a sharper question, which cases behave as neatly as the two known ones, and let the mathematics tell her the answer. She also kept going through years without a post. Choose a precise question, and keep working when the path is blocked.
Practice routine:
Compute the ratio of axial to diameter moment of inertia for a ring using NCERT Table 6.1.
Solve three angular momentum problems where you must first decide the axis.
Write one sentence for each of the Euler, Lagrange and Kovalevskaya cases describing where the centre of gravity sits.
Next step: take the disc in the worked example and redo it for a ring of the same mass and radius.
