-Relating Conservative Force and Potential

Energy

-Energy diagrams and Equilibrium

AP Physics CMrs. Coyle

A Force is “Conservative” if:

• “ the work this force does on an object that moves between two points depends only on the position of these two points and not on the path.”

• “the work this force does on an object that moves through a round trip is zero.”

• Example: gravity, force of a spring.

Consider a ball thrown up and the system is the ball

and the earth.

• The work done by the force of gravity Wg =-ΔU

For a particle moving along the x-axis (one dimensional motion)while a conservative

force, in the same axis, within the system acts on it:

f

i

x

C xxW F dx U ΔU is the change in potential

energy of the system

Solving for Fx:

x

dUF

dx

Example: Find the gravitational force for a

particle a distance x above the earth’s surface.

• Start with U= mgx

• Find F

Example: Find the spring force for a particle attached

to a spring:

• Start with U= ½ kx2

• Find F(x):

Types of Positions of Equilibrium

• Stable Equilibrium: movement away from this (x=0) position results in a restoring force.

• Unstable Equilibrium: movement away from this position results in a force directed away from x=0

• Neutral equilibrium: movement away from x=0 does not result in either restoring nor disruptive forces.

Energy Diagrams, U(x) vs x

-Energy Diagrams -U(x) is minimum at x=0

(stable equilibrium)

xmax and –xmax :

turning points

Unstable equilibriumU(x) is max

Neutral EquilibriumU(x) is constant

Example 1 (#40)A single conservative force acting on a

particle varies as F=(-Ax+Bx2 )i N, where A and B are constants and x is in meters.

a) Calculate the potential energy function taking U=0 at x=0

b) Find the change in potential energy and change in kinetic energy as the particle moves from x=2.00m to x=3.00m

Example 2 (#42)

• A potential energy function for a two dimensional force is U=3x3y - 7x

• Find the force that acts at the point (x, y)

• Hint find F(x) and F(y).