Final answer:
To find the rod's angular speed after the gum sticks, apply conservation of angular momentum using the initial momentum of the gum and the combined moment of inertia of the rod and gum system.
Step-by-step explanation:
To solve for the angular speed of the rod after the gum sticks to it, we can use the conservation of angular momentum because no external torques are acting on the system. Initially, the rod is at rest, so its angular momentum is zero. When the gum, which can be considered a point mass, sticks to the rod, the momentum of the gum becomes the angular momentum of the rod-gum system.
We can calculate the initial linear momentum of the gum and then equate it to the angular momentum of the combined system (L = Iω), where I is the moment of inertia and ω is the angular speed. The moment of inertia of the rod is I_rod = (1/12)m_rodL^2, and for the gum stuck at distance r from the center, it's I_gum = m_gumr^2.
The total moment of inertia is I_total = I_rod + I_gum. The initial angular momentum, L_initial, is the mass of the gum times its speed times the distance from the rotation axis (L_initial = m_gum * v_gum * r). The final angular momentum, L_final, is I_total * ω. Setting them equal (L_initial = L_final) and solving for ω gives us the angular speed of the system.