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Energy Using Calculus

Energy Using Calculus. If you work against a force like gravity (lifting) or a spring (stretching) W =F Δ x = Δ _?_. U. Where U stands for potential energy. Rearranging F = Δ U/ Δ x. or in the language of Calculus. A Force is “Conservative” if:.

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Energy Using Calculus

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  1. Energy Using Calculus

  2. 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

  3. 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.

  4. Consider a ball thrown up and the system is the ball and the earth. • The work done by the force of gravity Wg =-ΔU

  5. 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: ΔU is the change in potential energy of the system

  6. Example: Find the gravitational force for a particle a distance x above the earth’s surface. • Start with U= mgx • Find F

  7. Example: Find the spring force for a particle attached to a spring: • Start with U= ½ kx2 • Find F(x):

  8. What physical system could energy functions like this describe?

  9. What physical system could energy functions like this describe?

  10. 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.

  11. -Energy Diagrams -U(x) is minimum at x=0 (stable equilibrium) xmaxand –xmax : turning points

  12. Unstable equilibriumU(x) is max

  13. Neutral Equilibriumis where U(x) is constant U x

  14. 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

  15. 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).

  16. If 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 ? • I If an object had a total energy of 10 joules , how fast would it be going at C?__________ at D? ________ At E? ___________

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