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A GIF of the described behavior

I've always thought it was weird that pencils act like this: if one pulls their finger along the side of a pencil until it touches the surface below, the pencil is launched in the opposite direction of the way that the finger moved. Why is this?

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The sequence of events is shown below.

enter image description here

Initially the pencil is propelled forward with speed $v_{\rm A}$ but has backspin $\omega_{\rm A}$ (anticlockwise rotation) so there is relative movement between the pencil and the surface as $v_{\rm A} \ne r\omega_{\rm A}$ where $r$ is the radius of the pencil.

A kinetic friction force acts which reduces the rotational speed of the pencil $\omega_{\rm B}$ until there is no rotation of the pencil $\omega_{\rm C}=0$ but the pencil is still moving forward $v_{\rm C}$.

The frictional force then starts the pencil rotating clockwise with increasing angular speed and eventually the no slip condition, $v_{\rm D} = r\omega_{\rm D}$, is reached.

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As the finger comes down the side of the pencil, two things happen:

A compression that imparts a horizontal force taking the pencil away from the finger

And a rotation that causes the pencil to rotate tending to bring the pencil back to the start point

These combine to define how far the pencil travels before it stops.

The use of spin can be seen on a snooker or billiards table : top, bottom and side...

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