# Modelling the movement and jumps of a chalk while drawing a dashed line on a blackboard

You probably know that if you try to draw a line using a piece of chalk on a blackboard , under some conditions (for example, $\alpha<\frac{\pi}{2}$ in the picture below) you will have a dashed line pattern instead of a continuous line.

My question is :

(how) Can you model this special movement of the chalk ,and specially , find the length of line segments and the distance between them(which are the visible characteristics of the motion)?

Note:gravity is present. (I think it affects the solution, at least in some models) The pattern: • This is a great question and I am genuinely surprised how it has not been answered(till now) in 5 years. – Tausif Hossain Aug 5 '18 at 19:26
• I'm not sure either @Tausif Hossain. I have heard it referenced before a couple of times. A lot of info about it is buried though, when I tried to to find the coefficient of chalk I only found papers on climbing! – 3141 Aug 5 '18 at 20:02
• I agree, had a similar experience. It is indeed very difficult to find proper data on the co-efficient of friction of chalk online, let alone for static or kinetic. But, the paper you've suggested is very helpful, good find. – Tausif Hossain Aug 5 '18 at 20:09
• Thanks, I wonder whether there is a paper that addresses the paper from a purely physics point of view, rather than engineering @Tausif Hossain. Perhaps the Cambridge experiment mentioned produced a paper? – 3141 Aug 5 '18 at 20:39

## 1 Answer

The phenomena that you are talking about is called Stick-slip. This short paper explains it in detail. To summarise it, there are two types of friction coefficients: the static co-efficient, which acts on static objects, and is what stops an object from being moved, and the kinetic coefficient, which resists moving objects. When the chalk is pressed into the board, it yields momentarily to the force (this yielding depends on different factors including the porosity of the chalk. However, when the force applied overcomes the static coefficient, the chalk skips forward at speed. However, the adhesion between molecules in the chalk and the board means that some parts of the chalk are torn off, resulting in the mark: chalk relies on friction to make a mark. On the other hand, when it is moving, the lower kinetic coefficient means that the chalk does not make a mark. Once the chalk comes to a rest again, the process repeats itself, leaving a dotted line.

The formula for friction is $f =\mu n$, where $\mu$ is the frictional coefficient.

Let $F$ be the force applied to the chalk.

The friction experienced by the chalk would be $F\sin(\alpha)\mu$, where $\mu$ is either the static or kinetic coefficient, depending on whether the chalk is moving or not. The length of the dashes would be related to factors like area of the chalk in contact with the board and the creep of the chalk, and the length of the gaps to $F-F\sin(\alpha)\mu$. In particular, the the acceleration of the chalk would be $(F-F\sin(\alpha)\mu)/M$, where $M$ is mass.

• Hi! Welcome to Physics Stack Exchange. Note that here it's generally accepted to use MathJax to type equations etc. It looks great and is easier to read and you can get in symbols like $\mu$. So I highly recommend giving this link a read. Learning it is easy and a worthy investment. math.meta.stackexchange.com/questions/5020/… – Tausif Hossain Aug 5 '18 at 19:20
• @TausifHossain Not only it is accepted, it is encouraged :-) – AccidentalFourierTransform Aug 5 '18 at 19:34
• Yes, indeed. I remember being told the exact thing when I first came here on this site and it's come a long way. I just learnt how the "\sin" makes a difference from just "sin", from you so thanks. – Tausif Hossain Aug 5 '18 at 19:38
• Thanks for the time taken to edit my post. Sorry for not using correct formatting this time. – 3141 Aug 5 '18 at 19:55