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Physics 218: Mechanics

In this lecture by Dr. Tatiana Erukhimova, we delve into the Brachistochrone problem, exploring the curve of fastest descent known as the cycloid. The session covers key concepts in potential energy diagrams for conservative forces, illustrating equilibrium points and the behavior of particles in different energy states. We will analyze how a particle moves under the influence of a conservative force and address questions regarding kinetic energy and stability of equilibrium. This lecture combines theory with practical scenarios to enhance understanding of mechanics.

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Physics 218: Mechanics

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  1. Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 24

  2. Water Slide Who will come first?

  3. The curve of fastest descent Cycloid Inverted cycloid: Brachistochrone http://curvebank.calstatela.edu/brach/brach.htm

  4. ENERGY DIAGRAMS

  5. Potential Energy Diagrams • For Conservative forces can draw energy diagrams • Equilibrium points • Motion will move “around” the equilibrium • If placed there with no energy, will just stay (no force)

  6. Stable vs. Unstable Equilibrium Points The force is zero at both maxima and minima but… • If I put a ball with no velocity there would it stay? • What if it had a little bit of velocity?

  7. A particle moves along the x-axis while acted on by a single conservative force parallel to the x-axis. The force corresponds to the potential-energy function graphed in the Figure. The particle is released from rest at point A. • What is the direction of the force on the particle when it is at point A? • At point B?

  8. c) At what value of x is the kinetic energy of the particle a maximum? d) What is the force on the particle when it is at point C?

  9. e) What is the largest value of x reached by the particle during its motion? f) What value or values of x correspond to points of stable equilibrium? g) Of unstable equilibrium?

  10. Quiz

  11. Quiz A block of mass m is attached to a vertical spring, spring constant k. The spring is unstretched at y=0. A If the spring is compressed an amount A and the block released from rest, how high will it go?

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