---
title: "Yo-Yo & Spool Direction Puzzle Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/spool-puzzle/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/spool-puzzle/"
last_modified: "2026-10-22"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## The string and ground compete through torque

**Background** Pulling the string to the right accelerates the center to the right. The string pulls off-center, though, so it also produces a torque, and the ground contact has to accommodate both effects at once.

**Why it matters** The equations for translation and rotation follow OpenStax University Physics, sections 10.7 and 11.1. The torque from the tangent string and the work at the contacts are derived here for this apparatus.

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 misconception** Pulling to the right always makes the spool roll right.

**Better mental model** The 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 misconception** At the critical angle, the spool always stays still.

**Better mental model** The 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 misconception** The pull's work is just the horizontal force times how far the center moves.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

Physical conclusion

The critical-angle formula predicts a reversal only while the spool rolls without slipping and stays on the ground. The plotted envelope checks both conditions.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)