A slippery pencil

Physics is great, because it studies everything, from the expansion of the Universe, to the chaotic movements of molecules. However, before we go that far, I'd like to show you how physics describes the simple falling of a pencil on a table, from a vertical position(θ is 0 at the beginning)

      First, I want you to do the experiment on different surfaces, like in the image above. If you did it right, you should have observed that on smooth surfaces, the head of the pencil touching the surface goes onward or backwards.

It already seems a little complicated. In the next section I will write the equations of motion at a arbitrarily chosen moment during the falling.I strongly urge you to make the calculation by yourself too, because the way I will wrote might not be clear enough.

From the conservation of energy:

Iω^2/2= Mgl ( 1-cosθ) /2
where I is the moment of inertia of the pencil about  the contact point;I=Ml^2/ 3

ω^2=3g(1-cosθ)/l

 the tangential acceleration,  ɛ=dω/dt, is ɛ=3g sinθ /2l

• the equation of motion on the vertical axis is:

Mg-N = Mω^2*cosθ *l/2  + ɛsinθ *l/2

=> N=Mg((3cosθ-1)/2)^2
  
N becomes 0 at cosθ=1/3=> θlim=70.5; that means that at 70.5 degrees the pencil loses contact with the table(that holds only if the pencil hadn't started to slop till then- the equations of motion would change)

• the equation of motion of the horizontal axis is:

The F force is sketched having a arbitrarily chosen direction, being equal to the contact force between the surface and the pencil. If the direction chosen is wrong, I will obtain a negative value of the force from the equations, so that will be just fine( I'll change the direction if so).

F= Mɛ *cosθ* l/2 - Mω^2 *sinθ *l/2

=> F= 0.75 sinθ( 3 cosθ - 2)

the condition for the slipping to start is F= μN, so at that moment we can write:

μ=F/N= 3sinθ( 3 cosθ -2 )/ (3cosθ -1)^2

In order to study properly the motion of the pencil, we should sketch a graph between θ and μ, and see when does the pencil starts to slide and how. However a representation with μ=f(θ), will do just fine. 


for arbitrarily chosen values of  θ varying between 0 and 70.5 degrees, I obtained:

θ         μ

0         0

10       0.1301

20       0.2539

30       0.3512

40       0.3411

50       -0.191

60       -5.196

70       -4042

From μ=f(θ) we can figure out that μ becomes negative at cosθ=2/3 , θ=48

at θ=36, μ=0.37, which is the maximum value obtained.

Next, I'll  sketch the graph and I'll explain the falling of the pencil using it.

• The first part of the graph( black part) is attained when the pencil falls on a smooth surface.  The values of μ are positive, F=μN is positive( N>0 on this interval), so the direction of the force is well drawn. If the direction of F is to the right( as in the drawing), it means that the pencil starts to slip backwards. The conclusion is that the pencil starts to move backwards only if μ=0.37.
• The second part of the graph( red part) cannot be attained, because the pencil already started to slip, so the solutions are not physical anymore.
• The third part ( blue part) is attained when the pencil falls on a rough surface. The values of μ are negative,  F=μN is negative( N>0 on this interval), so the direction of the force F must be drawn in the other direction. The conclusion is that when 0.37<μ<1 the pencil starts moving onward.

Well, that's pretty much it...I leave you the pleasure to study the motion of the pencil after it started slipping. Maybe I'll study it in my future posts....