Energy Using

Calculus

If you work against a force like gravity (lifting) or a spring (stretching)

W =F Δx = Δ _?_

x

dUF

dx

UWhere U stands for potential energyRearranging

F = ΔU/Δxor in the language of Calculus

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

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):

What physical system could energy functions like this describe?

What physical system could energy functions like this describe?

Types 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) is minimum at x=0 (stable

equilibrium)

xmax and –xmax :

turning points

Unstable equilibriumU(x) is max

Neutral Equilibriumis where U(x) is constant

U

x

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).

If

• I

If an object had a kinetic energy of 2 joules at C, between what 2 points could it oscillate?

How much kinetic energy would it need to escape from this force ?

If an object had a total energy of 10 joules , how fast would it be going at C?__________ at D? ________

At E? ___________