Question

what is the angular velocity ω of the stick puck combination after the collision? assume that the stick is uniform and has a moment of inertia i0 about its center.

Answer 1

Using conservation of angular momentum to solve for the angular velocity after the collision, and the concepts of linear and rotational kinetic energy to calculate the kinetic energy before and after the collision.

Step-by-step explanation:

To solve for the angular velocity (ω) after the collision, we use the principle of conservation of angular momentum. The initial angular momentum of the system is simply the linear momentum of the small particle times the radius of the cylinder because it acts at a perpendicular distance from the axis of rotation. The formula for angular momentum (L) is given by L = r × p, where r is the radius, and p is the linear momentum of the particle.

Initially, the particle of mass m has a linear momentum given by p = mv, where v is its velocity. Therefore, L = r × mv. After the collision, the total angular momentum is conserved, but now the system consists of the combined particle and cylinder rotating with an angular velocity, which we denote ω. The moment of inertia of the cylinder is I0 and that of the particle at the edge of the cylinder is
`mr^(2)`. Therefore, the final angular momentum is (I0 +
`mr^(2)`)ω. Equating the angular momentums before and after the collision gives us: r × mv = (I0 +
`mr^2`)ω. From this equation, we can isolate ω and find the angular velocity of the system after the collision.

To calculate kinetic energy loss, we first determine the initial kinetic energy of the particle, which is ½
`mv^(2)`, and then the final kinetic energy of the system after the collision using the rotational kinetic energy formula ½ I0ω2 plus the kinetic energy of the rotating particle ½
`mr^2`ω2. The difference between the initial and final kinetic energies gives us the loss of kinetic energy due to the collision.