Pulling on a Spool

Physics Classical Mechanics • Physics of Rigid Bodies

Written by STEM Calculators Team Published May 9, 2026 Updated May 9, 2026

Analyze the classic spool or yo-yo paradox: depending on the inner radius, outer radius, and pull angle, the spool may roll toward the pull, roll away from the pull, or stay at the critical condition. The calculator uses \[ a=\frac{FR(R\cos\theta-r)}{I+MR^2}, \qquad \alpha=\frac{a}{R}. \]

Preset

Spool inertia mode

Spool mass M

Mass unit

Outer rolling radius R

Inner string radius r

Length unit

Pull force F

Force unit

Pull angle θ

Available static friction μₛ

Gravity g

Custom inertia and preview options

Positive acceleration means the spool rolls toward the pull direction. Negative acceleration means it rolls away. The direction flips at \[ \theta_c=\cos^{-1}\!\left(\frac{r}{R}\right). \]

Custom inertia factor k

Moment of inertia I

I unit

Preview time t

Energy output

Diagram zoom

Animation speed

With rightward pull as positive, the rolling direction is controlled by \[ R\cos\theta-r. \] If \(R\cos\theta>r\), the spool rolls toward the pull. If \(R\cos\theta

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Enter spool geometry, pull angle, and force, then click “Calculate”.

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Theory – Pulling on a spool

Pulling on a spool is a classic rolling-motion paradox. Even though the string is pulled to the right, the spool can roll to the right, roll to the left, or sit at the critical transition depending on the geometry and pull angle.

1. Geometry of the problem

The spool has an outer rolling radius \(R\) and an inner string radius \(r\). A force \(F\) pulls the string at an angle \(\theta\) above the horizontal.

The key geometric quantity is

\[ R\cos\theta-r. \]

This quantity determines whether the torque about the contact point tends to roll the spool toward or away from the pull.

2. Critical angle

The transition occurs when the line of action gives zero torque about the contact point:

\[ R\cos\theta_c-r=0. \]

Therefore,

\[ R\cos\theta_c=r. \]

Solving for the critical angle gives

\[ \theta_c=\cos^{-1}\!\left(\frac{r}{R}\right). \]

If \(\theta<\theta_c\), then \(R\cos\theta>r\), and the spool accelerates toward the pull. If \(\theta>\theta_c\), then \(R\cos\theta

3. Rolling direction rule

Condition	Result	Meaning
\(R\cos\theta>r\)	Rolls toward the pull	The torque about the contact point drives the center of mass in the pull direction.
\(R\cos\theta=r\)	Critical condition	The translational acceleration is zero in the ideal rolling model.
\(R\cos\theta	Rolls away from the pull	The pull torque dominates in the opposite rolling sense.

4. Torque about the contact point

The easiest way to predict the acceleration is to take torques about the point of contact with the ground. For rolling without slipping, the spool has an effective moment of inertia about the contact point

\[ I_P=I+MR^2. \]

The torque of the pull about the contact point is

\[ \tau_P=F(R\cos\theta-r). \]

Therefore,

\[ \alpha=\frac{\tau_P}{I_P} = \frac{F(R\cos\theta-r)}{I+MR^2}. \]

With the rolling condition \(a=\alpha R\),

\[ a= \frac{FR(R\cos\theta-r)}{I+MR^2}. \]

5. Inertia models

The calculator can use a direct value of \(I\), or an approximation of the form

\[ I=kMR^2. \]

Model	Moment of inertia	Use case
Direct input	\(I=\text{given}\)	Best when the problem gives the spool’s exact moment of inertia.
Solid disk approximation	\(I=\frac12MR^2\)	Useful as a simple rough model.
Hoop approximation	\(I=MR^2\)	Useful when most mass is near the outer rim.
Custom factor	\(I=kMR^2\)	Useful for realistic spools with mass between the hub and the rim.

6. Static friction force

Static friction is not what causes the direction paradox by itself. Instead, it adjusts to enforce rolling without slipping. In the horizontal direction,

\[ F\cos\theta+f_s=Ma. \]

Thus,

\[ f_s=Ma-F\cos\theta. \]

If \(f_s>0\), friction acts to the right. If \(f_s<0\), friction acts to the left.

7. Static friction requirement

The upward component of the pull reduces the normal force:

\[ N=Mg-F\sin\theta. \]

The maximum available static friction is

\[ f_{s,\max}=\mu_sN. \]

Rolling without slipping is possible only if

\[ |f_s|\leq \mu_sN. \]

If \(F\sin\theta>Mg\), the spool loses contact with the surface, so the rolling-on-ground model no longer applies.

8. Worked example

A spool has inner radius

\[ r=3\ \mathrm{cm}, \]

outer radius

\[ R=8\ \mathrm{cm}, \]

and the string is pulled at

\[ \theta=40^\circ. \]

First find the critical angle:

\[ \theta_c=\cos^{-1}\!\left(\frac{r}{R}\right). \] \[ \theta_c= \cos^{-1}\!\left(\frac{3}{8}\right). \] \[ \theta_c\approx68.0^\circ. \]

Since

\[ 40^\circ<68.0^\circ, \]

the spool rolls toward the pull.

The same result follows from checking

\[ R\cos\theta-r. \] \[ R\cos40^\circ-r = 8\cos40^\circ-3 \approx3.13\ \mathrm{cm}. \]

This is positive, so the acceleration is in the pull direction.

Key idea: the spool’s direction is determined by the sign of \(R\cos\theta-r\), not simply by the direction of the applied force.

Frequently Asked Questions

Why can a spool roll toward or away from the pull?

The pull creates both a horizontal force and a torque through the inner radius. The net rolling tendency depends on the sign of R cos(theta) - r.

What is the critical angle for pulling a spool?

The critical angle is theta_c = arccos(r/R). At this angle, the ideal translational acceleration is zero.

When does the spool roll toward the pull?

The spool rolls toward the pull when R cos(theta) is greater than r, or equivalently when theta is less than theta_c.

When does the spool roll away from the pull?

The spool rolls away from the pull when R cos(theta) is less than r, or equivalently when theta is greater than theta_c.

What formula is used for acceleration?

Using torque about the contact point, a = F R (R cos(theta) - r) / (I + M R^2).

How is angular acceleration found?

For rolling without slipping, alpha = a / R, with the sign indicating the rolling direction.

What does static friction do in this problem?

Static friction adjusts to enforce rolling without slipping. Its value is found from F cos(theta) + f_s = M a.

How do you check if rolling without slipping is possible?

Compute the required friction f_s and compare it with the maximum available static friction mu_s N. Rolling without slipping requires |f_s| <= mu_s N.

What happens if the upward pull is too large?

If F sin(theta) is greater than Mg, the normal force becomes zero or negative and the spool loses contact with the surface.

For r = 3 cm, R = 8 cm, and theta = 40 degrees, which way does the spool roll?

The critical angle is arccos(3/8), about 68 degrees. Since 40 degrees is below this, the spool rolls toward the pull.

Pulling on a Spool

Custom inertia and preview options

Pull angle, critical angle, and rolling direction

Calculation steps

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Frequently Asked Questions