Definition of a “Conservative Force” The work done by a “conservative force” on a particle moving between two points ______________________ on the path taken by the particle. Is a gravitational force conservative? ___ Is a spring force conservative? ___ Is friction a conservative force? ___ The work done by gravity to move an object from point 1 to point 2 = _____, so the work done is _________ for the three different paths. does NOT depend YES mgh h the same YES NO

Potential Energy Curves If a particle is part of a system in which a conservative force acts, then there is a _____________________ function associated with that force. When only a conservative force acts, the mechanical energy, E, of the system _____________. We can learn about the motion of the particle from a GRAPH of the Potential Energy function, U(x)… potential energy is constant (conserved)

Example 1 : A bead on a wire moves under the influence of gravity only (NO friction!) Shape of wire Potential Energy Curve Why is the graph of U(x) the same curve as the shape of the wire? U=mgh, which is a linear function (As h ↑,U ↑)

Describe the resulting motion of the bead in each case: E = U + K If E = 5 J, the particle can NOT move to the left of ___, because the particle can NOT have _________ kinetic energy. ___ is called a ________ point. There is ___ ________________ on the right. x1x1 negative x1x1 turning NO turning point

E = U + K If E = 4 J, turning point between ___ and ___, & _________ if x > x 5. This is called _______ equilibrium (like a marble on a flat table) If E = 3.5 J, ________________________ ______________________________. x1x1 x2x2 stationary neutral turning points on each side Bead would oscillate forever!

If E = 3 J, and bead is placed at x 3, K = ___. This is called ________ equilibrium (like a marble balanced on top of a bowling ball.) If E = 1 J, the bead must be at ___, and, again, K = ___. This is called ________ equilibrium (like a marble placed at the bottom of a curved bowl). E = U + K 0 unstable x2x2 0 stable (like the “balancing bear”)

Example 2 : Sketch the potential energy curve for a system consisting of a mass on a spring. X = 0 X U(x)

Example 3 : The potential energy of two atoms in a molecule is given by: where x = the distance between atoms in nm Find the equilibrium separation, x o, between the atoms (the point where the system is at its lowest potential energy) At x o, U(x) is at a __________, so the ______ (or __________ of the function) at that point is ______. minimum slopederivative zero

Calculate the minimum potential energy of the system, which occurs at the equilibrium separation, x o, so find ______

Plot the conservative force, F(x) acting between the two atoms. Before we attempt this, what is the relationship between F(x) and U(x)? and In the differential limit:

So… the force at any point is the _________ of the ______ of the potential energy curve at that point. F(x) x xoxo F = _____ at x o oppositeslope zero + F = _______ force repulsive