To solve this problem, we need to analyze the collision between the particle and the rod using principles of conservation of momentum and rotational dynamics. Let's break it down step by step.
1. Angular Velocity of the Rod After Impact
When the particle of mass 'm' strikes the end B of the rod at a right angle, it exerts an impulse on the rod. Since the collision is perfectly elastic, both momentum and kinetic energy are conserved. The rod is hinged at point A, which means it can rotate about this point.
First, we need to find the angular momentum of the system about point A before and after the collision. The initial angular momentum (L_initial) of the system can be calculated as:
Here, 'l' is the distance from point A to point B, and 'u' is the velocity of the particle.
After the collision, let the angular velocity of the rod be 'ω'. The moment of inertia (I) of the rod about point A is given by:
Using the conservation of angular momentum:
- L_initial = L_final
- m * u * l = I * ω
Substituting the moment of inertia into the equation:
- m * u * l = (1/3) * M * l² * ω
Now, solving for ω:
- ω = (3 * m * u) / (M * l)
2. Velocity of the Particle After Impact
In a perfectly elastic collision, both momentum and kinetic energy are conserved. We can use the conservation of linear momentum in the horizontal direction (assuming the rod is initially at rest):
Here, 'v' is the velocity of the rod's end B after the collision, and 'v'' is the velocity of the particle after the collision.
From the angular velocity we found earlier, the velocity of point B after the impact can be expressed as:
- v = ω * l = (3 * m * u) / (M * l) * l = (3 * m * u) / M
Now substituting this back into the momentum equation:
- m * u = m * v' + M * (3 * m * u) / M
- m * u = m * v' + 3 * m * u
Rearranging gives:
- m * v' = m * u - 3 * m * u
- v' = -2 * u
3. Impulsive Reaction at End A of the Rod
The impulsive reaction at the hinge A can be found by analyzing the forces acting on the rod during the collision. The impulse at A must balance the change in momentum of the rod and the particle. The total impulse (J) can be expressed as:
Substituting the value of 'v' we found earlier:
- J = M * (3 * m * u) / M - m * u = 3 * m * u - m * u = 2 * m * u
The impulsive reaction at point A is equal to the change in momentum of the rod and can be expressed as:
Where Δt is the duration of the collision. Since we are interested in the impulsive reaction, we can express it as:
In summary, we have:
- Angular velocity of the rod after impact: ω = (3 * m * u) / (M * l)
- Velocity of the particle after impact: v' = -2 * u
- Impulsive reaction at end A of the rod: R = 2 * m * u / Δt
This analysis provides a comprehensive understanding of the dynamics involved in the collision between the particle and the rod. Each step relies on fundamental principles of physics, ensuring that the results are consistent with the laws of motion and energy conservation.
