Presentation on theme: "PH 201 Dr. Cecilia Vogel Lecture 20. Potential Energy Graph It is very often useful to look at a graph of potential energy as a function of position."— Presentation transcript:
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.
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.
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!
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
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)
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)
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)
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.
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.
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)
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
Stable and Unstable Equilibrium If you displace it just a bit what happens?