Energy
Using
Calculus
If you work against a force like gravity
(lifting) or a spring (stretching)
W =F Δx = Δ _?_
U
Where U stands for potential energy
Rearranging
F = ΔU/Δx
or in the language of Calculus
dU
Fx  
dx
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:
xf
WC   Fx dx  U
xi
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 equilibrium
U(x) is max
Neutral Equilibrium
is 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
•
If an object had a kinetic
energy of 2 joules at C,
between what 2 points
I 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? ___________