# Determination of the speed and angular velocity of 4 rolling cylinders between planks

According to the picture, there are four identical cylinders, with radius R, that are rolling without slipping at every contact point. The upper plank is moving with speed 2v (m/s) towards the right, while the lower plank is moving with speed v (m/s) towards the left. My question here is, what is the speed at point P? It seems intuitive to conclude that Vp = 2v but if we ignore the lower plank and cylinders, shouldn't the displacement of the 1st upper cylinder + displacement of the 2nd upper cylinder = displacement of the upper plank? I'm pretty confused at this part. Also in this configuration is the velocity of the center of mass of each cylinder not equal to (angular velocity)*R anymore?

Starting with a lower cylinder (and assuming that the given v and 2v are measured relative to the base), it is clear that the center of mass is moving to the left with a speed of v/2. Combine that with a tangential velocity of v/2 to get v at the top and an angular velocity of v/(2R). For an upper cylinder $$v_c$$ + $$v_t$$ = 2v and $$v_c$$ - $$v_t$$ = -v. (Where $$v_c$$ is the speed of the cylinder to the right, and $$v_t$$ is the tangential speed.