M053 · Why a spool can roll the wrong way

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.

Look around a three-dimensional spool whose string leaves the inner hub along a tangent. Follow marks on the turning rim and on the moving ground, look at the forces where the string pulls and where the spool meets the floor, and check that the work done by the pull equals the kinetic energy plus the heat at the contact. Adjust the pull angle, hub radius, mass distribution and grip. Compare underhand and overhand winding, trace how the initial acceleration changes with the pull angle, and compare how far the center moves with how far the rim turns. At the critical angle the spool stays still because the forces balance.

Angular acceleration $0\,\mathrm{rad\,s^{-2}}$

Net torque about center $0\,\mathrm{N\,m}$

Contact heating rate $0\,\mathrm{W}$

String material velocity along pull $0\,\mathrm{m\,s^{-1}}$

Recorded time $0\,\mathrm{s}$

Calculated normal force $0\,N$

Inertia divided by mass and squared radius $0$

Underhand candidate reversal angle $0\,{}^{\circ}$

Current center acceleration $0\,\mathrm{m\,s^{-2}}$

Center displacement $0\,m$

Center speed $0\,\mathrm{m\,s^{-1}}$

Counterclockwise angular speed $0\,\mathrm{rad\,s^{-1}}$

Bottom contact velocity $0\,\mathrm{m\,s^{-1}}$

Signed contact friction $0\,N$

Required rolling friction magnitude $0\,N$

Available static friction $0\,N$

Actual kinetic coefficient $0$

Total kinetic energy $0\,J$

Integrated contact heat $0\,J$

Pull work at moving string contact $0\,J$

Work-energy-heat balance defect $0\,J$

Predict the motion, then test your design

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

Forces at their application pointsWeight is blue, support violet, friction red and string tension amber, all at one force scale per trial.Bottom material-point velocityThe teal arrow shows the signed slip relative to the surface, which is zero in pure rolling. Velocity arrows use their own scale.See through the front flangeThe transparent cutaway reveals the tangent string and the contact forces; the mass and inertia keep their full values.Follow a material point on the rimThe white marker turns with the recorded body angle. The violet hub and the string have no mass.Body mass

m

2.00\,\mathrm{kg}

Outer rolling radius

R

0.30\,\mathrm{m}

String hub radius ratio

r/R

0.55\,

Body hole radius ratio

R_{\mathrm{in}}/R

0.20\,

Pull angle above horizontal

\alpha

25.00\,{}^{\circ}

Constant string tension

F

5.00\,\mathrm{N}

Static friction coefficient

\mu_s

0.60\,

Kinetic-to-static coefficient ratio

\mu_k/\mu_s

0.70\,

Winding: positive underhand, negative overhand

w

1.00\,

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Drag the hollow purple hub marker to change the string's lever arm. The blue curve is the center displacement and the dashed purple 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 ring with a massless hub and a massless, inextensible string. Rolling resistance and air drag are ignored. The pull keeps the same angle and strength for four seconds. If the calculated normal force is zero or negative, the ground-contact model is dropped, and any flight that follows is outside this apparatus. Rolling and sliding follow exact solutions for constant loads and are checked by an independent balance of work, kinetic energy and heat. The reversal angle applies only to underhand winding with enough static friction.

Start with the essentials

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