Aim: How do we analyze energy diagrams and stability of equilibrium?

Do Now: Do Now What force function is associated with the following potential energy function, U(r) = 5r^3 -7r? a)F(r)=0 b)F(r)= 15r^2 c) F(r)=15r^2-7 d)F(r)= -15r^2 e) F(r)= -15r^-2+7

Potential Energy Curve of a Spring Mass System

Think Pair and Share: Determine at which position is the magnitude of the force the greatest A is the greatest force

Equilibrium Types

Stable Equilibrium The positions of stable equilibrium correspond to those values of x for which U(x) has a relative minimum value on an energy diagram.

Energy Diagram The potential energy is graphed. The dashed line represents the total energy of the particle. Therefore, the kinetic energy at a given point is equal to the difference between the total energy and potential energy at that point.

Writing Assignment: Analyze the potential energy Curve Find a position of stable equilibrium. In 1-2 complete full sentences, explain what will happen to the particle if it is released near this point. C

How can we tell this is stable equilibrium? In stable equilibrium, the force is always opposite the displacement from a stable point. (If x>0, F is negative and if x<0, F is positive) Determine the direction of the force at point –A. To the right Determine the direction of the force at point +A To the left

Unstable Equilibrium Positions of unstable equilibrium correspond to those values of x for which U(x) has a relative maximum value on an energy diagram. For any finite displacement from a position of unstable equilibrium, the particle accelerates away from the point.

Writing Assignment: Unstable Equilibrium Determine a point of unstable equilibrium. E In 1-2 complete sentences, explain what will happen if the particle is released near this point.

Analyze the motion by looking at the potential energy curve Describe the motion of the particle if it is released from rest at point B. The particle oscillates between points B and D

Problem 1: Analyze the potential energy curve on the following slide At each point in the curve, state whether the force is positive, negative, or neutral. A=-,B=0,C=+,D=0,E=-,F=0 b) At which point does the force have the greatest magnitude? C c) Write down the letter which corresponds to stable equilibrium, unstable equilibrium, and neutral equilibrium. B=Unstable Equilibrium F=Neutral Equilibrium D= Stable Equilibrium

Problem 2: Analyze the potential curve on the following slide Rank regions AB, BC, CD, and DE according to the magnitude of the force on the particle, greatest first. AB>CD>BC=DE What value must the mechanical energy Emec of the particle not exceed if the particle is to be trapped in the potential well BC? 5J What value must the mechanical energy Emech of the particle not exceed if the particle is to be trapped in potential well FG? 5J

Problem 2: Analyze the Potential energy curve on the following slide d) What energy would allow the particle to move between the two potential wells but not to the right of point H? 6J