A thin rod has a length of 0.339 m and rotates in a circle on a frictionless tabletop. The axis is perpendicular to the length of the rod at one of its ends. The rod has an angular velocity of 0.687 rad/s and a moment of inertia of 1.03 x 10-3 kg·m2. A bug standing on the axis decides to crawl out to the other end of the rod. When the bug (whose mass is 5 x 10-3 kg) gets where it's going, what is the change in the angular velocity of the rod?

To find the change in angular velocity of the rod when the bug moves from one end to the other, we can make use of the principle of conservation of angular momentum.

The angular momentum of the system (rod plus bug) is conserved when there are no external torques acting on the system. In this case, since the tabletop is frictionless, we can assume that there are no external torques.

The angular momentum (L) of an object is given by the product of its moment of inertia (I) and its angular velocity (ω): L = I * ω.

Initially, the angular momentum of the system (L1) is the product of the moment of inertia of the rod (I1) and its initial angular velocity (ω1). When the bug moves to the other end of the rod, the moment of inertia of the system changes to I2 (including the moment of inertia of the bug). The final angular velocity of the system is ω2.

Since angular momentum is conserved, we have L1 = L2, which implies:
I1 * ω1 = I2 * ω2

To find the change in angular velocity (Δω), we rearrange the equation as follows:
Δω = ω2 - ω1 = (I1 * ω1) / I2 - ω1

Now, let's plug in the given values:
I1 = 1.03 x 10⁻³ kg·m² (moment of inertia of the rod)
ω1 = 0.687 rad/s (initial angular velocity of the rod)
I2 = I1 + mb * L² (moment of inertia of the system after the bug moves, including the bug's moment of inertia)
where mb = 5 x 10⁻³ kg (mass of the bug)
L = 0.339 m (length of the rod)

Substituting the values into the equation, we have:
I2 = (1.03 x 10⁻³ kg·m²) + (5 x 10⁻³ kg) * (0.339 m)²
I2 ≈ 1.03 x 10⁻³ kg·m² + 5.46 x 10⁻⁵ kg·m²
I2 ≈ 1.08546 x 10⁻³ kg·m²

Now, we can calculate the change in angular velocity (Δω):
Δω = (I1 * ω1) / I2 - ω1
Δω = (1.03 x 10⁻³ kg·m² * 0.687 rad/s) / (1.08546 x 10⁻³ kg·m²) - 0.687 rad/s

Evaluating this expression, we find:
Δω ≈ 0.409 rad/s

Therefore, the change in the angular velocity of the rod when the bug moves from one end to the other is approximately 0.409 rad/s.