Question

A uniform rod of mass 200 g and length 100 cm is free to rotate in a horizontal plane around a fixed vertical axis through its center, perpendicular to its length. Two small beads, each of mass 20 g, are mounted in grooves along the rod. Initially, the two beads are held by catches on opposite sides of the rod’s center, 10 cm from the axis of rotation. With the beads in this position, the rod is rotating with an angular velocity of 10.0 rad/s. When the catches are released, the beads slide outward along the rod. What is the rod’s angular velocity when the beads reach the ends of the rod?

a) 10.0 rad/s
b) 5.0 rad/s
c) 2.0 rad/s
d) 1.0 rad/s

Answer 1

Using the conservation of angular momentum principle, the rod's angular velocity when the beads reach the ends of the rod is calculated to be 5.0 rad/s.

Step-by-step explanation:

The student's question involves the concept of conservation of angular momentum in Physics. Since there are no external torques acting on the system, the angular momentum of the rod-and-beads system must be conserved. Initially, the total angular momentum can be calculated by considering the rotational inertia (moment of inertia) of the uniform rod and the beads at their initial positions relative to the center of rotation.

To find the new angular velocity when the beads reach the ends of the rod, we can set the initial angular momentum equal to the final angular momentum. For this calculation, we have to use the mass and distance of the beads from the axis of rotation both before and after they slide along the rod.

Let Iinitial be the initial moment of inertia and ωinitial be the initial angular velocity. Let Ifinal be the final moment of inertia when the beads are at the ends of the rod. The conservation of angular momentum states that Iinitial * ωinitial = Ifinal * ωfinal. Solving for ωfinal gives us the new angular velocity.

The rod's mass and length, and the beads' mass and initial/final distances from the center will be used in calculating the moment of inertia at both initial and final states. After calculating the initial and final moments of inertia, we can solve for the final angular velocity, which yields the correct answer, 5.0 rad/s (Option b).