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?

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I'm just trying to look at the answers? How did i get here?

To determine the change in the angular velocity of the rod when the bug crawls to the other end, we need to use the principle of conservation of angular momentum.

The angular momentum of a system remains constant unless an external torque is applied. In this case, the bug crawling from one end to the other will exert a torque on the system, causing a change in angular momentum.

Angular momentum (L) is given by the product of moment of inertia (I) and angular velocity (ω):

L = I * ω

The moment of inertia of the rod is already given as 1.03 x 10^(-3) kg·m^2, and the initial angular velocity is 0.687 rad/s. Therefore, the initial angular momentum (L_initial) of the rod-bug system can be calculated as:

L_initial = I * ω_initial

Next, when the bug reaches the other end of the rod, the moment of inertia changes because the mass distribution of the system changes. To calculate the final moment of inertia (I_final), we need to consider both the rotating rod and the added mass of the bug:

I_final = I_rod + I_bug

The rod is thin and uniform, so we can use the formula for the moment of inertia of a thin rod rotating about its end:

I_rod = (1/3) * m * L^2

Where m is the mass of the rod and L is the length of the rod. Given the length of the rod as 0.339 m, we can calculate I_rod.

Finally, knowing the mass of the bug (5 x 10^(-3) kg), we can calculate the moment of inertia of the bug (I_bug) using the formula:

I_bug = m_bug * r^2

Where r is the distance of the bug from the axis of rotation.

Now, we have the initial angular momentum (L_initial) and the final moment of inertia (I_final). By conserving angular momentum, we can solve for the final angular velocity (ω_final) of the system using the equation:

L_initial = I_final * ω_final

Rearranging the equation to solve for ω_final gives us the change in angular velocity (Δω):

Δω = ω_final - ω_initial = L_initial / I_final - ω_initial

By plugging in the values obtained from the calculations, we can find the change in the angular velocity of the rod when the bug reaches the other end.