Yo-Yo & Spool Direction Puzzle

Does a spool always roll the way you pull its string? Not always; the angle of the pull matters. With the string coming off the underside of the hub, a shallow pull to the right rolls the spool to the right, and a pull steeper than a critical angle rolls it left. The critical angle is where the string points straight at the spot where the spool touches the ground. This holds while the spool rolls without slipping.

Drag the string angle or hub radius to reverse a spool. Compare rolling, sliding, center travel and pull work in a live 3D apparatus.

Does a spool always roll the way you pull its string?

Body mass

m

2\,\mathrm{kg}

Outer rolling radius

R

0.3\,\mathrm{m}

String hub radius ratio

r/R

0.55

Body hole radius ratio

R_{\mathrm{in}}/R

0.2

Pull angle above horizontal

\alpha

25\,\mathrm{{}^\circ}

Constant string tension

F

5\,\mathrm{N}

Static friction coefficient

\mu_s

0.6

Kinetic-to-static coefficient ratio

\mu_k/\mu_s

0.7

Winding: positive underhand, negative overhand

w

\text{Underhand}

Inspect recorded time

t/t_{\mathrm{end}}

22\,\%

Drag the violet hub handle to change the string's lever arm. The teal curve is the center displacement and the dashed violet curve is the rim rotation with its sign reversed; the two coincide only in pure rolling. The envelope shows the initial acceleration and has a gap where ground contact is lost. Because the hub is massless, the winding and the body's density can be changed independently.

The spool is a rigid uniform annulus; its hub and inextensible string are massless. Rolling resistance and air drag are ignored. Pull angle and tension stay constant for four model seconds. A nonpositive calculated normal force invalidates the ground-contact solution; subsequent flight is outside this model. Constant-load rolling and sliding use exact solutions checked against work, kinetic energy and frictional heat. The reversal angle is a geometric candidate for underhand winding and requires sufficient static friction. The camera follows the center; floor marks move with the true displacement. The front flange is translucent to expose the hub. Geometry, arrows and the freely extended tangent string are schematic.

Focus question: When can a pull to the right produce motion to the left?
One-sentence intuition: With underhand winding, the string turns the spool counterclockwise. Rolling to the right needs a clockwise angular acceleration, so ground friction can make the center accelerate the other way.

Core mathematical model

Forces and torque with signs

X=F\cos\alpha,\quad \tau=wFr,\quad N=mg-F\sin\alpha,\quad w\in\{-1,+1\}

A winding sign of plus one means underhand and minus one means overhand. If the normal force is zero or negative, the ground model no longer applies.

Rolling acceleration and friction

a=\frac{F(\cos\alpha-wr/R)}{m(1+\beta)},\quad f_{\mathrm{req}}=-\beta ma-\frac\tau R,\quad\beta=\frac I{mR^2}

Use this acceleration only if the normal force is positive and static friction can supply the required force.

Angle at which the direction can reverse

\alpha_c=\arccos(r/R)\quad(w=+1),\qquad |f_{\mathrm{req}}|\le\mu_sN

Above this angle, an underhand spool with enough grip rolls left. Without enough grip, the spool need not stay at rest at this angle.

Work at the string contact

W=X\Delta x+\tau\Delta\theta,\qquad W=\Delta K+Q,\qquad Q=\int\mu_kN|v+R\omega|\,\mathrm{d}t

The string's work includes a rotational term. In pure rolling, the static ground contact does no work; sliding turns energy into heat.

Common difficulties

The pull direction is not enough

Typical misconceptionPulling to the right always makes the spool roll right.

Better mental modelThe pull also exerts a torque, and the ground contact has to accommodate both the push and the twist.

The critical angle works only under conditions

Typical misconceptionAt the critical angle, the spool always stays still.

Better mental modelThe static friction needed may exceed its limit, or the string may lift the spool off the ground.

Work depends on where the force acts

Typical misconceptionThe pull's work is just the horizontal force times how far the center moves.

Better mental modelThe force acts off-center, at the tangent point on the hub, so it also does work as the spool turns.

Run the experiment

01
Check the tangent before the torque
Use the string-and-contact camera, then switch the winding and change the hub radius.
What to observe: Where the string leaves the hub, it stays perpendicular to the radius. Switching the winding reverses the torque, even though the pull still points right.
02
Compare power at the string
Complete a trial with slipping and compare the string speed, the contact heating and the energy balance.
What to observe: The speed of the string itself combines translation and rotation. The total work equals the kinetic energy plus the heat at the contact.
03
Predict a reversal
Compare the Underhand: roll right and Underhand: roll left presets.
What to observe: The acceleration envelope changes sign within the range where rolling holds.
04
Test exact equilibrium
Select Exact critical angle and complete the pull trial.
What to observe: Friction at the contact holds the spool still, balancing both the force and the torque.
05
Take away the grip
At the critical angle, reduce static friction to almost zero.
What to observe: The spool slips, and the reversal prediction based on rolling no longer applies.
06
Change winding or lose contact
Try Overhand winding, then Loss of ground contact.
What to observe: With overhand winding there is no reversal angle in this range. Once the spool lifts off, the ground model no longer applies.

Core mathematical model

Forces and torque with signs

Rolling acceleration and friction

Angle at which the direction can reverse

Work at the string contact

Common difficulties

The pull direction is not enough

The critical angle works only under conditions

Work depends on where the force acts

Run the experiment

Check the tangent before the torque

Compare power at the string

Predict a reversal

Test exact equilibrium

Take away the grip

Change winding or lose contact