Фізика Йо-йо: Обертальний Момент Інерції та Струна, Якої Неможливо Розв’язати

Two kinds of kinetic energy from one fall

Drop an ordinary ball and every joule of its gravitational potential energy becomes translational kinetic energy, 1/2 m v^2. A yo-yo unwinding on its string does something different: as it falls a height h, the released energy mgh has to supply both the translational kinetic energy of its center of mass falling, 1/2 m v^2, and the rotational kinetic energy of its spin about its own axle, 1/2 I w^2. That split, and only that split, is why a yo-yo descends noticeably slower than free fall even though gravity is the only external force doing net work on it.

Rolling without slipping on the string: v = wr

The string does not unwind from the yo-yo's outer rim — it unwinds from the narrow axle at radius r, which is much smaller than the disk's outer radius R. Because the string does not slip against the axle, the linear speed of the falling center of mass and the angular speed of the spin are locked together by v = w r, exactly analogous to a wheel rolling without slipping. This constraint is what couples the two energy terms into a single, solvable equation of motion instead of two independent ones.

Moment of inertia sets the acceleration

Modelling the yo-yo as a uniform disk gives I = 1/2 m R^2 about its own axis; a real yo-yo, with more mass concentrated near the rim, behaves like I = k m R^2 for some shape factor k somewhat above 1/2. Applying Newton's second law separately to the translational and rotational motion and eliminating the string tension gives a downward acceleration far smaller than g whenever the axle radius r is small compared with R, since a small r makes the moment-of-inertia term dominate the denominator.

translation:  mg - T = m a
rotation:     T * r = I * alpha,   with a = alpha * r

solving:      a = m g r^2 / (I + m r^2) = g / (1 + I / (m r^2))

small axle radius r  ->  I/(m r^2) large  ->  a much less than g

The 'sleeping' bottom and the snap-up

When the string finally runs out, the yo-yo is spinning fast, but the friction between string and axle is far too weak to stop that spin instantly — instantly halting the rotation would require a huge string tension the friction simply cannot supply. So the string goes slack and the yo-yo keeps spinning freely on the axle, or "sleeps." A small upward tug on the string re-engages the axle's grip (the deliberate "bind" technique used with modern unresponsive yo-yos), catching the string again and letting the stored angular momentum pull the yo-yo back up the string — though friction losses mean it never quite reaches its starting height, and every up-down cycle loses a little more energy.

Why a thinner axle makes a 'better' yo-yo

A narrower axle radius r increases the ratio I / (m r^2) in the acceleration formula above, pushing more of the released potential energy into spin rather than downward speed, and stretching out the sleep time at the bottom. Modern competitive yo-yos push this further with ball-bearing axles, which cut friction dramatically and let the yo-yo sleep for tens of seconds — at the cost of needing that deliberate bind to return it to the hand, since the near-frictionless axle no longer grips the string on its own the way a classic fixed axle did.

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