Mechanics problem: rod rolling of a step

[rayf]

Hello i am having trouble understanding what would be the angular and linear velocity of the center of mass of a rod that falls of a step and hits the ground and continues rolling. I know that mechanical energy is not conserved after the rod hits the ground, as that would be an inelastic collision, also angular momentum is not conserved as there is friction acting to rotate the rod once it hits the floor. I am also aware that the rod will roll and move while slipping till v=wr but how can i determine an equation so that i can know when one velocity will decrease and the other increase until they are equal.

 

[LightHero]

The angular and linear velocity of the center of mass of a rod that falls of a step and hits the ground and continues rolling can be determined by using the equations of motion. The equation of motion for the angular velocity is ω = \frac{v}{r}, where v is the linear velocity and r is the radius of the rod. The equation of motion for the linear velocity is v = at, where a is the acceleration due to gravity and t is the time elapsed since the rod started rolling. The two equations can be combined to give \omega = \frac{at}{r}.As the rod rolls, friction will act to decrease both the angular and linear velocities. The magnitude of the friction force acting on the rod depends on the coefficient of friction between the rod and the surface it is rolling on. The angular velocity will decrease more rapidly than the linear velocity as the rod moves, until they are equal when v = wr. At this point, the rod will stop rolling and slip until it comes to a complete stop.