Final answer:
The question involves determining the acceleration of a rod when a particle is projected and then subsequent angular speed after collision, using principles like Newton's laws, conservation of momentum, and conservation of angular momentum. Due to a lack of specific values, calculations cannot be completed, but the equations based on physical laws can be stated.
Step-by-step explanation:
The problem presents a situation with a uniform rod and a particle connected by a string, describing two sequential physical events: the projection of the particle and its subsequent collision with the rod. Such problems are typically solved using principles from mechanics, including Newton's laws, conservation of momentum, and rotational dynamics.
Acceleration of the Rod's Center
Upon the particle's projection, the center of the rod experiences an acceleration due to the tension in the string. Since the particle applies a force equal to its mass times acceleration (F=ma), and using Newton's second law, we can set up an equation for the acceleration of the rod's center, taking into account the mass of the rod and the mass of the particle. However, without specific values or further details, the exact numerical acceleration cannot be calculated.
Angular Speed After Collision
After the particle collides with the center of the rod and sticks to it, we must use the conservation of angular momentum to find the post-collision angular speed. Before the collision, the particle has linear momentum, which is converted into angular momentum when it sticks to the rod. Since no external torques are acting on the system, the angular momentum is conserved, and thus we can equate the initial and final angular momentum to solve for the final angular speed. The equation involves the mass of the particle, the rod, the length of the rod, and the initial velocity of the particle.