12. Dotted line trick

Official Problem Statement

“Under certain conditions, when a piece of chalk is dragged across a blackboard, the resulting trace is not a continuous line but a periodic sequence of dots, even though the motion is uninterrupted. This can happen with other writing materials and surfaces. Investigate the physical parameters that govern this phenomenon.”

The Stick-Slip Mechanism

Rather than sliding uniformly, chalk on a blackboard undergoes stick-slip oscillation. The root cause is the velocity-weakening friction force (the Stribeck effect): static friction $f_s$ exceeds kinetic friction $f_k$, and $f_k$ decreases with increasing sliding speed. This creates a negative-damping regime that makes uniform sliding unstable.

Simple model: chalk (effective mass $m$, effective stiffness $k_{\rm eff}$) is dragged at constant hand speed $v_0$.

Stick phase: chalk is in contact with the board, elastic load builds up:

$$m\ddot{x} + k_{\rm eff}(x – v_0 t) = 0$$

When the elastic force reaches $f_s$, chalk suddenly slips.

Slip phase: chalk detaches or lifts slightly from the surface (friction drops to $f_k \ll f_s$), executes free vibration, and leaves no mark:

$$m\ddot{x} + k_{\rm eff}(x – v_0 t) = -f_k \cdot \text{sgn}(\dot{x} – v_0)$$

The oscillation frequency during stick-slip is:

$$f_{\rm osc} \approx \frac{1}{2\pi}\sqrt{\frac{k_{\rm eff}}{m}}$$

Dot Spacing and Governing Parameters

The oscillation period is $T = 1/f_{\rm osc}$. In each period the chalk advances by $d_{\rm dot}$:

$$\boxed{d_{\rm dot} = v_0 \cdot T = \frac{v_0}{f_{\rm osc}} = \frac{2\pi v_0}{\sqrt{k_{\rm eff}/m}}}$$

The dot spacing is therefore proportional to the dragging speed $v_0$ and inversely proportional to the square root of the effective stiffness $k_{\rm eff}$.

Critical Speed

Below a critical dragging speed $v_c$, stick-slip does not develop and the chalk leaves a continuous line. The critical speed follows from the stability condition of the equilibrium sliding state:

$$v_c \approx \sqrt{\frac{k_{\rm eff}}{m}} \cdot \frac{f_s – f_k}{(\partial f/\partial v)\big|_{v=0}}$$

Above $v_c$, the stick-slip cycle becomes self-sustaining (a stable limit cycle in the phase plane), analogous to Helmholtz motion of a bowed string or squealing brakes.

Effective Stiffness

The effective stiffness $k_{\rm eff}$ combines the chalk’s own stiffness, the flexural stiffness of the hand and arm, and the contact stiffness of the surface:

$$\frac{1}{k_{\rm eff}} = \frac{1}{k_{\rm chalk}} + \frac{1}{k_{\rm arm}} + \frac{1}{k_{\rm contact}}$$

Parameters

Experimental Setup

To measure $d_{\rm dot}$ as a function of $v_0$ and $\theta$:

1. Mount chalk on a linear rail driven by a stepper motor at precisely controlled speeds.
2. Scan or photograph the resulting trace and measure dot spacing by image processing.
3. Plot $d_{\rm dot}$ versus $v_0$ — expect a linear relationship with slope $1/f_{\rm osc}$.
4. Repeat at different inclination angles $\theta$ and with added masses to vary $m$ independently.
5. Attach a small accelerometer to the chalk to directly measure $f_{\rm osc}$ and compare to $\sqrt{k_{\rm eff}/m}/(2\pi)$.

References

• Rabinowicz, E., Friction and Wear of Materials, 2nd ed., Wiley, 1995 — Velocity-dependent friction.
• Leine, R. I., van Campen, D. H., de Kraker, A., “Stick-Slip Vibrations Induced by Alternate Friction Models,” Nonlinear Dynamics 16 (1998) 41–54.
• Akay, A., “Acoustics of friction,” J. Acoust. Soc. Am. 111 (2002) 1525–1548.
• Stoimenov, B. L. et al., “The roughness effect on the frequency of frictional sound,” Tribol. Int. 40 (2007) 659–664.