A rod of mass 12m and length L lying on a smooth horizontal plane can rotate freely about a stationary vertical axis passing through the Rod`s end. A particle of mass m moving along the plane with speed v hits the rod end perpendicularly and gets stuck there. Find the angular velocity of system.

To solve this problem, we need to apply the principle of conservation of angular momentum. When the particle collides with the rod and sticks to it, the total angular momentum of the system before the collision must equal the total angular momentum after the collision, assuming no external torques are acting on the system.

Understanding the System Before the Collision

Initially, we have a rod of mass 12m and length L that can rotate about one of its ends. The particle of mass m is moving with speed v and strikes the rod perpendicularly at its end. Before the collision, the rod is at rest, so its angular momentum is zero. The particle, however, has angular momentum relative to the axis of rotation.

Calculating Initial Angular Momentum

The angular momentum (L) of the particle before the collision can be calculated using the formula:

• L = r × p

Where:

• r is the distance from the axis of rotation to the point of impact (which is L, the length of the rod).
• p is the linear momentum of the particle, given by p = mv.

Thus, the initial angular momentum of the particle is:

• L_initial = L × mv

After the Collision

Once the particle sticks to the end of the rod, we need to consider the combined system of the rod and the particle. The total moment of inertia (I) of the system can be calculated by adding the moment of inertia of the rod and the moment of inertia of the particle.

Calculating the Moment of Inertia

The moment of inertia of the rod about one end is given by:

• I_rod = (1/3) * (12m) * L² = 4mL²

The moment of inertia of the particle, which is treated as a point mass at a distance L from the axis, is:

• I_particle = m * L²

Therefore, the total moment of inertia of the system after the collision is:

• I_total = I_rod + I_particle = 4mL² + mL² = 5mL²

Applying Conservation of Angular Momentum

According to the conservation of angular momentum:

• L_initial = L_final

Where L_final is the angular momentum of the combined system after the collision, which can be expressed as:

• L_final = I_total * ω

Here, ω is the angular velocity of the system after the collision. Setting the initial and final angular momentum equal gives us:

• L × mv = (5mL²) * ω

Solving for Angular Velocity

Now we can solve for ω:

• ω = (L × mv) / (5mL²)

Notice that the mass m cancels out:

• ω = (Lv) / (5L²) = v / (5L)

Final Result

The angular velocity of the system after the particle collides with the rod and sticks to it is:

• ω = v / (5L)

This result shows how the initial linear momentum of the particle translates into angular motion of the combined system after the collision, illustrating the principles of rotational dynamics and conservation laws effectively.