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Sandbox Physics

M053 · Contact / direction reversal

Yo-Yo & Spool Direction Puzzle

Pull a string tangent to an inner hub and tune its angle, hub radius, mass distribution and grip. Compare underhand and overhand winding, trace the initial-acceleration envelope and watch center displacement against rim rotation. The critical-angle case is a force balance, not a frozen animation.

Interactive modelYo-Yo & Spool Direction Puzzle
Recorded time0 s0\,\mathrm{s}
Calculated normal force0 N0\,N
Inertia divided by mass and squared radius00
Underhand candidate reversal angle0 ∘0\,{}^{\circ}
Current center acceleration0 m s−20\,\mathrm{m\,s^{-2}}
Center displacement0 m0\,m
Center speed0 m s−10\,\mathrm{m\,s^{-1}}
Counterclockwise angular speed0 rad s−10\,\mathrm{rad\,s^{-1}}
Bottom contact velocity0 m s−10\,\mathrm{m\,s^{-1}}
Signed contact friction0 N0\,N
Required rolling friction magnitude0 N0\,N
Available static friction0 N0\,N
Actual kinetic coefficient00
Total kinetic energy0 J0\,J
Integrated contact heat0 J0\,J
Pull work at moving string contact0 J0\,J
Work-energy-heat balance defect0 J0\,J

Physics tutorial

The string and ground compete through torque

BackgroundA rightward pull accelerates the center directly, but its off-center point of application also supplies a torque. The ground must reconcile both tendencies.

Why it mattersThe translational and rotational balances follow OpenStax University Physics sections 10.7 and 11.1. The tangent-string torque and contact-work expressions are derived for the apparatus here.

Start with the essentials

Focus question
When can a pull to the right produce motion to the left?
One-sentence intuition
Underhand winding supplies counterclockwise string torque. Rolling right requires clockwise acceleration, so ground friction can reverse the center response.

Core mathematical model

Signed loads

X=Fcos⁡α,τ=wFr,N=mg−Fsin⁡α,w∈{−1,+1}X=F\cos\alpha,\quad \tau=wFr,\quad N=mg-F\sin\alpha,\quad w\in\{-1,+1\}

Positive winding is underhand; negative winding is overhand. A nonpositive normal force invalidates the ground model.

Rolling acceleration and friction

a=F(cos⁡α−wr/R)m(1+β),freq=−βma−τR,β=ImR2a=\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.

Candidate reversal angle

αc=arccos⁡(r/R)(w=+1),∣freq∣≤μsN\alpha_c=\arccos(r/R)\quad(w=+1),\qquad |f_{\mathrm{req}}|\le\mu_sN

Above this angle an adequately gripped underhand spool rolls left. With insufficient grip, the candidate angle need not produce rest.

Work at the string contact

W=XΔx+τΔθ,W=ΔK+Q,Q=∫μkN∣v+Rω∣ dtW=X\Delta x+\tau\Delta\theta,\qquad W=\Delta K+Q,\qquad Q=\int\mu_kN|v+R\omega|\,\mathrm{d}t

String work includes the rotational term. Static ground contact does no work in pure rolling; sliding produces heat.

Common difficulties

Pull direction is insufficient

Typical misconceptionA rightward force guarantees rightward rolling.

Better mental modelBoth translation and torque must satisfy the ground contact constraint.

Critical angle is conditional

Typical misconceptionThe candidate critical angle always freezes the spool.

Better mental modelThe necessary static friction may exceed its limit, or the string may lift the body.

Work follows the applied point

Typical misconceptionPull work is only horizontal force times center displacement.

Better mental modelThe off-center force also does rotational work through the tangent hub.

Run the experiment

  1. 01

    Predict a reversal

    Compare underhand right and underhand left presets.

    What to observe: The envelope changes sign inside the valid rolling region.
  2. 02

    Test exact equilibrium

    Select the critical-angle preset and complete the pull trial.

    What to observe: The supplied contact friction balances both translation and rotation.
  3. 03

    Remove enough grip

    At the critical angle, reduce static friction to almost zero.

    What to observe: The body slips and the rolling reversal prediction no longer applies.
  4. 04

    Change winding or lose contact

    Try overhand winding, then loss of ground contact.

    What to observe: Overhand has no reversal angle in this range; lift is explicitly outside the ground model.