Download - 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):
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.
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? ___________