What is the value of vp for a particle ejected from a rotating rod?

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SUMMARY

The discussion focuses on calculating the speed of a particle ejected from a rotating rod. The rod has a length of 0.790 m and rotates at an angular speed of 22.0 rad/s. The particle, with a mass of M/3.00, is ejected with a speed (vp) that is 6.00 m/s greater than the rod's end speed at the moment of ejection. By applying the conservation of angular momentum, the relationship between the rod's moment of inertia (1/12ML²) and the particle's angular momentum is established as crucial for solving the problem.

PREREQUISITES
  • Understanding of angular momentum and its conservation principles
  • Familiarity with the moment of inertia, specifically for a rod (1/12ML²)
  • Knowledge of rotational dynamics and linear speed relationships
  • Basic algebra for solving equations involving angular and linear quantities
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  • Study the principles of conservation of angular momentum in rotational systems
  • Learn how to calculate the moment of inertia for various shapes
  • Explore the relationship between linear speed and angular speed in rotating objects
  • Practice solving problems involving ejected particles from rotating bodies
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The figure below is of an overhead view of a thin uniform rod of length 0.790 m and mass M rotating horizontally at angular speed 22.0 rad/s about an axis through its center. A particle of mass M/3.00 attached to one end is ejected from the rod and travels along a path that is perpendicular to the rod at the instant of ejection. If the particle's speed vp is 6.00 m/s greater than the speed of the end of the rod just after ejection, what is the value of vp?
W0245-N.jpg





Li=Lf, L=Iw, l=rmv



The Attempt at a Solution


- I haven't been able to attempt this problem because I'm having problems setting it up. I'm trying to set up the initial side right now and I am using 1/12ML^2 for the rod, but unsure what to use for the particle since it's inside of the rod. I think I can solve the problem if I can get past this step.
 
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Use conservation of angular momentum. The angular momentum of the particle about the center is mvr
 

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