Final answer:
To find the angular velocity of the system after the collision, we can apply the principle of conservation of angular momentum. The small particle sticks to the edge of the uniform solid cylinder, causing the cylinder to start rotating. The final angular velocity of the system depends on the mass and distribution of the small particle.
Step-by-step explanation:
To find the angular velocity of the system after the collision, we can apply the principle of conservation of angular momentum. Before the collision, the small particle has a linear momentum given by its mass (m) multiplied by its velocity (v). The solid cylinder is initially at rest, so its initial angular momentum is zero. After the collision, the system will have a combined angular momentum equal to the sum of the angular momenta of the small particle and the solid cylinder.
The angular momentum (L) of an object is given by the product of its moment of inertia (I) and its angular velocity (ω). The moment of inertia of the small particle is negligible compared to that of the solid cylinder, so we only need to consider the angular momentum of the cylinder. The moment of inertia of a solid cylinder rotating about its central axis is given by the equation I = (1/2) * M * R^2, where M is the mass of the cylinder and R is its radius.
Using the principle of conservation of angular momentum, we can set the initial angular momentum of the system equal to the final angular momentum of the system:
L_initial = L_final
0 + (1/2) * M * R^2 * 0 = (1/2) * M * R^2 * ω_final
Since the small particle sticks to the edge of the cylinder, the final angular velocity of the system will be determined by the moment of inertia and mass distribution of the combined system. Therefore, the mass and distribution of the small particle will affect the final angular velocity of the system.