# 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

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