Falling stick problem (no friction): What is the kinetic energy?

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SUMMARY

The discussion focuses on the kinetic energy of a falling stick with no friction, emphasizing the equation for kinetic energy as the sum of rotational and translational components. The correct expression for kinetic energy is established as \( K.E = \frac{1}{2} I_{cm} \dot{\alpha}^2 + \frac{1}{2} m v_{cm}^2 \), where \( I_{cm} \) is the moment of inertia about the center of mass, \( \dot{\alpha} \) is the angular velocity, and \( v_{cm} \) is the velocity of the center of mass. The conversation highlights the importance of considering both rotational and translational kinetic energy in the absence of friction and energy dissipation.

PREREQUISITES
  • Understanding of rotational dynamics and kinetic energy equations
  • Familiarity with the concepts of moment of inertia and angular velocity
  • Knowledge of the principles of conservation of energy in physics
  • Basic grasp of Newton's laws of motion
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  • Study the derivation of kinetic energy equations in rotational motion
  • Learn about the moment of inertia for various shapes and its significance
  • Explore the effects of conservative forces on energy conservation
  • Investigate the role of friction and air resistance in energy dissipation
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Physics students, educators, and anyone interested in understanding the dynamics of rotational motion and energy conservation principles in mechanics.

phos19
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Since there is no friction : $$ m \ddot{x} = 0 $$ (no x motion).For the kinetic energy , I've tried: $$ K = 1/2 I_{cm} \dot{\alpha}^2 + 1/2 m v^2_cm = 1/2 I_{cm} \dot{\alpha}^2 + 1/2 m \dot{z}^2$$ . Giving me a weird expression , shouldn't the kinetic energy just be half the the moment of inertia about the contact point times $$\dot{\alpha}^2$$?
 
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K.E would be sum of rotational and translational kinetic energy. Gravity is a conservative force, and if there's no air resistance and assuming that no energy dissipation happens due to heat and sound etc, then K.E of falling stick = Rotational K.E about it's Center of mass + Translational K.E of the center of mass.
 
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