# Relative motion - Cylinders rolling between two identical planks

Today i ran onto this simple problem, which seemed to be interesting to me. Given the illustration bellow, the problem is states as:

Two identical cylinders roll in between two identical planks. If the velocity of each cylinder is $\vec{v}$ and the velocity of the bottom plank is $\vec{u}$ ($|\vec{v}| > |\vec{u}|$), find the velocity the man standing on the top plank needs to obtain with respect to it[the top plank], for which he covers distance $s$ in $t$ seconds in the stationary reference frame ( observer's frame ). Will the velocity vector point right or left? ( $\vec{v}$ and $\vec{u}$ are both given with respect to the stationary reference frame ). All friction is to be neglected and the cylinders are assumed to be rotating without slipping. Let $\vec{v_m}$ be the sought velocity.

Interestingly, a naive approach would be to immediately write $$\vec{v_m} = \frac{\vec{s}}{t} - \vec{v}$$ since one might think that the velocity of the upper plank with respect to the stationary observer is$\vec{v}$.

However, after inspecting the problem more thoroughly, I've come to the following conclusion:

• The velocity of the upper plank comes solely from the rotation of the cylinders.
• Angular velocity of each of the cylinders is$\$ $\omega = \frac{v-u}{R}$

From here it seems that the velocity of the upper plank with respect to the stationary reference frame might be $$\vec{u}' =\vec{v} - \vec{u}$$

And the sought velocity is: $$\vec{v_m} = \frac{\vec{s}}{t} - \vec{u'}$$

Which seems to be ok intuitvely ( the greater the magnitude of $\vec{v}$, the smaller the relative speed of the man needs to be ). However, it's still bugging me and making me believe the motion of the bottom plank affects the motion of the upper plank ( other than in the elaborated way ). Is there something wrong with this reasoning?

Note: Although this is not a school problem, I'll tag it as homework because this seems the kind of a problem that would appear in homework problems.

EDIT: By "stationary reference frame" I mean the frame on which the observer is standing ( e.g. Earth ), observing motion of the system"

## 1 Answer

When a cylinder is rolling, the relative velocity between its bottom and top is twice velocity of COM. In the given problem, if you take u=0, then you can see that the velocity of upper plank with respect to lower plank is 2v. Therefore, when the plank moves in the same direction as cylinder, its velocity gets added to this quantity giving you the velocity of upper plank with respect to stationary plane as 2v+u.

Now, this is the velocity of the man when he is standing still on the plank. For him to move s units in t seconds in the same direction as plank and cylinder , he needs a velocity of s/t in the same direction. So now you need to subtract the velocity of still man ( 2v+u) from s/t. This will give you the necessary velocity in the direction of motion of cylinder and planks. If it is positive, then the man needs to move faster in this direction as he is moving too slow and covering less than s units in t seconds . If it is negative, then he needs to run in the opposite direction as he is moving too fast and covering more than s units in t seconds.