Translational vs rotational momentum

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SUMMARY

The discussion focuses on the dynamics of an elastic collision between a point particle and a thin rod in space, emphasizing the transfer of momentum. Key equations include conservation of linear momentum (mv = mv1 + Mv2) and angular momentum (mvL/2 = mv1*L/2 + Iw, where I = mL^2 / 12). The analysis leads to the derivation of the ratios Mv2/mv and Iw/mv, illustrating how momentum is divided between translational and rotational forms post-collision. This exploration is crucial for understanding momentum conservation in isolated systems.

PREREQUISITES
  • Understanding of elastic collisions and momentum conservation principles
  • Familiarity with linear and angular momentum equations
  • Knowledge of moment of inertia calculations
  • Basic grasp of physics concepts related to motion in space
NEXT STEPS
  • Study the derivation of moment of inertia for various shapes
  • Learn about the principles of elastic and inelastic collisions
  • Explore advanced topics in rotational dynamics and torque
  • Investigate real-world applications of momentum conservation in engineering
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Physics students, educators, and professionals in engineering or mechanics who seek to deepen their understanding of momentum transfer in collisions and rotational dynamics.

Yaridovich
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Suppose I am floating in space and have a thin rod oriented vertically floating in front of me. This rod has its center of mass in the center of the rod (uniform mass distribution) and is not fixed to anything. I then throw a point particle with non-negligible mass horizontally;it hits the rod near one of its ends and an elastic collision occurs. How much of the point particle's momentum is transferred to the rod as translational momentum as opposed to angular momentum?
 
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particle - mass m, rod - mass M, length L,

before collision, particle velocity v. rod at rest
after collision, particle velocity v1, rod - translational velocity v2, rotational velocity w.

conserve linear & angular momentum, & energy,

linear momentum : mv = mv1 + Mv2

angular momentum : mvL/2 = mv1*L/2 + Iw. I = mL^2 / 12

energy : mv^2 = mv1^2 + Mv2^2 + Iw^2.

elimiate v1 & calculate Mv2 / mv and Iw / mv
 

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