Final answer:
The linear velocity of the center of the rod when its top end touches the floor is √(2gL), corresponding to option A.
Step-by-step explanation:
When the rod falls freely to one side without slipping, it undergoes both translational and rotational motion. To find the linear velocity of the center of the rod when its top end touches the floor, we can use the principle of conservation of energy.
At the top, the potential energy is given by mgh, where m is the mass of the rod and h is the height from the bottom end to the top end. At the bottom, the potential energy is zero, but the rod has gained kinetic energy. This kinetic energy is given by (1/2)mv^2, where v is the linear velocity of the center of the rod.
Setting the potential energy equal to the kinetic energy, we have mgh = (1/2)mv^2. Solving for v, we get v = sqrt(2gh).
Since the length of the rod is L, we can substitute h=L into the equation. Therefore, the linear velocity of the center of the rod when its top end touches the floor is √(2gL), which corresponds to option A.