About the Yo-Yo Physics Simulation
This simulation models a yo-yo as a solid disk with moment of inertia I = ½MR² that unrolls from a thin axle of radius r. As the string unwinds, gravity competes with string tension to set the rate of descent. Newton's second law for translation and rotation combine to give the constant downward acceleration a = g / (1 + R²/(2r²)), which the animation integrates step by step using the canvas.
The sliders let you set the mass M (20–300 g), outer radius R (15–55 mm), axle radius r (2–15 mm), string length (20–100 cm) and bearing losses (0–20%). Watching gravitational potential energy split into translational and rotational kinetic energy makes the device a classic teaching example of rotational dynamics, conservation of energy and how a small axle "gears down" the fall.
Frequently Asked Questions
Why does a yo-yo fall so slowly instead of dropping freely?
Because the string forces the yo-yo to spin as it falls, gravity must share its energy between linear motion and rotation. The acceleration is a = g / (1 + R²/(2r²)). Since the outer radius R is much larger than the axle radius r, the denominator is large, so a is only a tiny fraction of g and the descent is gentle.
What is the moment of inertia used in this model?
The yo-yo is treated as a uniform solid disk, so its moment of inertia about the central axis is I = ½MR², where M is the mass and R is the outer radius. This is computed live from the mass and outer-radius sliders and feeds directly into the acceleration and the rotational kinetic energy ½Iω².
What do the five sliders control?
Mass M sets the disk's mass in grams, outer radius R and axle radius r set the disk and axle sizes in millimetres, string length sets the drop distance in centimetres, and bearing losses add a small velocity-dependent drag in percent. Mass cancels out of the acceleration, but it scales every energy value and the bearing loss governs how quickly the yo-yo eventually stops.
How is energy distributed during the fall?
On the way down, gravitational potential energy converts into translational kinetic energy (½Mv²) and rotational kinetic energy (½Iω²). Because v = ωr is fixed by the string, the split between the two stays in the constant ratio R²:r². The coloured bars in the panel show these three energy stores updating in real time.
Why does the axle radius matter so much?
The axle radius r appears squared in the acceleration formula and sets how fast the disk spins for a given fall speed, since ω = v/r. A thinner axle makes the yo-yo spin faster, stores more energy in rotation and slows the descent; a thicker axle lets it fall more quickly. This is exactly the "gearing" trick real yo-yo designers exploit.
What happens when the yo-yo reaches the bottom?
When the string is fully unwound the yo-yo "sleeps" momentarily, then the spinning disk rewinds the string and climbs back up. In the simulation this is handled as a reversal of velocity with a small loss applied, so each rise reaches slightly less than the previous drop until the motion dies away.
Is this simulation physically accurate?
The core physics is sound: it uses the correct solid-disk inertia, the standard acceleration formula and proper energy bookkeeping. It is an idealisation, however—it ignores string stretch and thickness, air resistance and the precise mechanics of the "sleep" and rewind, instead modelling losses as a simple tunable percentage at each turning point.
Does the mass of the yo-yo change how fast it falls?
No. Mass cancels in the acceleration a = g / (1 + R²/(2r²)), so a light and a heavy yo-yo of the same shape descend at the same rate, just as in free fall. Mass does increase every energy reading and gives a heavier yo-yo more momentum, helping it spin or "sleep" for longer in practice.
What does the bearing-losses slider represent?
It models friction at the axle and air drag as an energy sink proportional to speed. At 0% the yo-yo bounces between top and bottom almost indefinitely; raising it removes energy on each pass so the oscillation decays and the device comes to rest sooner, mimicking a real yo-yo eventually stopping.
How does this relate to real yo-yo tricks like "the sleeper"?
A sleeper is a yo-yo spinning freely at the bottom of a fully unwound string. The more rotational kinetic energy ½Iω² it carries, the longer it sleeps before friction stops it. Players choose heavier rims and low-friction bearings to maximise that stored spin—the same quantities the rotational-KE bar in this simulation tracks.
How are the equations solved over time?
The simulation steps forward with a fixed acceleration and updates velocity, position and rotation angle each animation frame using a small time step (capped at about 33 ms). The angle advances by ω = v/r, and at each top or bottom the velocity is reversed with the chosen loss applied. This numerical integration is what produces the smooth, continuously updating motion you see.