# What determines the speed required to pull a table cloth?

I was watching this show "Street Genius" on National Geographic and the host Tim Shaw demonstrated an experiment about Inertia, What he did was, He tied one end of a table cloth to a car through a long rope and started driving the car, when the rope was taut, the cloth was pulled and the things on the table were still on the table. What are the calculations involved in it and what determines the speed required to pull the table cloth without disturbing the things on it?

• I guess the entire problem depends on friction between different objects in the table and the cloth, which adds complexity to the problem. – Waffle's Crazy Peanut Apr 19 '14 at 15:00
• It has no relation with speed... only acceleration – evil999man Apr 19 '14 at 16:05
• – Qmechanic May 2 '17 at 19:51

Not sure anyone will look back at this, but I'd like to give an answer anyway!

How do you not disturb the dishes when pulling a tablecloth out from under them? You're exactly right: this is about the inertia of the dishes and the forces on the dishes from the table cloth while the cloth is being pulled.

Remember from physics that if we plot the velocity of the dish vs time, then the slope is the acceleration, and the area under the curve is the distance traveled. We want the distance traveled to be less than the distance to the table edge.

To have a small area under the curve, then we want a small amount of time that the dish is contacting the table cloth (t*) and also a small effective friction force (the initial slope). So either pull fast or use a slippery table cloth.

The effective friction force is a tricky thing to predict, since the dish will surely rattle around and only be in contact part of the time. In fact, predictions about even simple friction coefficients are nearly impossible without just doing the experiment (see citation).

Nature 430, 525-528 (29 July 2004) | doi:10.1038/nature02750; The nonlinear nature of friction: Michael Urbakh1, Joseph Klafter1, Delphine Gourdon & Jacob Israelachvili • The important thing here is that the friction forces go like $\mu F_N$ for some $\mu$, hence the acceleration of the plates is approximately constant, as your "mountain" diagram shows. The total distance moved by those plates is $\frac 12 a t^2$, so you just need to minimize $t$. The answer is therefore the faster the better; the distance the plates travel must go like $\frac 12 \mu g (L/v)^2$ where $L$ is the length of the tablecloth, $\mu$ is the coefficient of kinetic friction, $g$ is the gravitational acceleration, and $v$ is the speed of the tablecloth/car. – CR Drost Jul 17 '15 at 16:10