1. PH 201 Dr. Cecilia Vogel Lecture 20

2. Potential Energy Graph  It is very often useful to look at a graph of potential energy as a function of position.  Why?  Can determine forces and subsequent motion of an object subjected to this potential energy.

3. U(x) Graph and Energy  If there are no external forces acting on the system, and no energy is dissipated, then  energy is conserved. K+U = E = constant  Where U is small, K = E-U is large  object moves fast.  Where U is large, but less than E, K is small  object moves slowly.

4. U(x) Graph and Energy  If energy is conserved, K+U = E = constant  Where U = E, K = 0  object stops (momentarily)  turning point  Where U > E, K = E-U  object cannot go there with that amount of energy!  “forbidden region”  (note — quantum) K can’t be negative!

5. Pendulum Example The potential energy of a pendulum can be plotted as a function of angle.  U = mgh  h = L-Lcos   U = mgL(1-cos  )  h L L cos 

6. Pendulum U(  ) U = mgL(1-cos  )   If it starts from rest at 10 o,  How much energy does it have?  Will it go right or left?  It can’t go right. U can’t be larger than 30J.  As it goes left, what happens to the speed?  AS U gets smaller, K gets bigger — moves faster. 10 o E = U i +K i = 30J 5o5o 0 10 20 30 40 -5 o -10 o U (J)

7. Pendulum U(  ) U = mgL(1-cos  )   Starts from rest at 10 o,  E=30J  As it goes left, moves faster.  Until…   =0, it goes fastest, K=30J  continues left (inertia)  As U gets bigger, K gets smaller, slows down.  Until… 10 o 5o5o 0 10 20 30 40 -5 o -10 o U (J)

8. Pendulum U(  ) U = mgL(1-cos  )  Starts from rest at 10 o, with E=30J  As it goes left, moves faster.  Until  =0, where it continues, but now slowing down,  until it stops at  = -10 o, where U=30J, K=0  This is turning point.  Then starts back right…  back and forth…  10 o 5o5o 0 10 20 30 40 -5 o -10 o U (J)

9. Graph Ordinate  Graph of U vs. x is not a picture of the motion.  The y-axis is potential energy, not height.  For example, for a mass on a spring sliding along a surface,  U= ½kx 2.  The graph looks like this, even though the mass goes along a line,  never up or down.

10. U(x) Graph and Forces  Consider the graphs of potential energy and force for gravity and a spring GRAVITYSPRING U= ½kx 2 x F= - kx x U= mgy y F= - mg y NOTE: F = opposite the slope of U graph.

11. Pendulum Forces  When the pendulum is at -5 o, and moving left (-)  since the slope is (-),  the force is (+),  so it is slowing down.  When the pendulum is at +5 o, and moving left (-)  since the slope is (+),  the force is (-),  so it is speeding up.  10 o 5o5o 0 10 20 30 40 -5 o -10 o U (J)

12. Stable and Unstable Equilibrium  Suppose the force  a potential  F = -d U /dx  Zero force implies  Unstable equilibrium means  if you displace it just a bit  it will move

13. Stable and Unstable Equilibrium  If you displace it just a bit  what happens?