Potential energy (chapter seven)

1. Potential energy (chapter seven) • Potential energy of a system • The isolated system • Conservative and nonconservative forces • Conservative forces and potential energy • The nonisolated system in steady state • Potential energy for gravitational and electrostatic forces • Energy diagrams and stability

2. Potential energy of a system • Objects and force internal to system • External work done on a system that does not change the kinetic or internal energy – change in potential energy • Gravitational potential energy - the work done by an external force raising an object from ya to yb is:

3. Potential energy This work is a transfer of energy to the system. We can define the quantity Ugmgy to be the gravitational potential energy The work done on the system then gives the change in Ug: W= Ug The change in energy is the key thing – to do problems, you need to first define a reference location (height) Only depends on height – not on horizontal displacement

4. Isolated systems Consider the system of an object only in the earth’s gravitational field, falling from yb to ya. In free fall, the work done by gravity is mg(yb-ya), which results in a change in the kinetic energy (work-kinetic energy theorem) K. This work equals -Ug, the change in the gravitational potential energy of the (earth + object) system. The earth’s kinetic energy will not change, so the change in kinetic energy of the (earth + object) system is just the change of the KE of the object. This gives the result: K= -Ug, or K+Ug=0

5. Isolated systems • We can write this as a continuity equation for the mechanical energy Emech=K+Ug  Emech=0, or Ki+Ui=Kf+Uf

6. Example – object in free fall • Consider earth and object as system • Object dropped from height y=h (y=0 is defined as height at which Ug=0) • At height y, what is the speed?

7. y=h Ug=mgh K=0 y Ug=mgy K=½mv2 y=0 Ug=0 Example – object in free fall Initial energy = K+Ug=mgh Final energy (at any point y) = mgy+½mv2 mgh=mgy+½mv2 v=(2g(h-y))½

8. Conceptest • Demo

9. m1  m2 Example If m1>m2, How fast will m1 be going when it hits the floor? Start: K+Ug=m1gh End: ½m1v2+m2gh m1gh=½m1v2+m2gh v=[2gh(m1-m2)/m1]½

10. Conservative and nonconservative forces Conservative forces • Forces internal to system that cause no transformation of mechanical to internal energy • Work done is path independent • Work done over closed path = 0 • Examples: gravitational, elastic

11. Conservative and nonconservative forces Potential energy of a spring Us½kx2 Gravitational potential energy Ugmgh New formulation of Work-KE thm: K+U+Eint=constant Conservation of energy

12. Example Motion on a curved track Child slides down an irregular frictionless track (total height h) , starting from rest. What is the speed at the bottom? Ki+Ui= Kf+Uf 0+mgh=½mv2+0  v=(2gh)½

13. Conservative forces and potential energy Since the work done by a conservative force can be written as W=-U We can express a differential amount of work done as dW=-dU=F·dr From this we can see that a conservative force can be written as Fx=-dU/dx (F=-U) e.g. Fg=-dUg/dy=-d(mgy)/dy=-mg

14. The nonisolated system in steady state Conservation of energy holds regardless of whether the system is isolated or not. For a nonisolated system, the net energy change can still be zero if the amount of energy entering equals the amount leaving the system.

15. Potential energy for gravitational and electrostatic forces Gravitational force between two masses (m1, m2) separated by a distance r This gives a general form for the gravitational potential energy of: And for the electrostatic potential energy

16. Example

17. Example

18. Energy diagrams and stability Since the potential energy associated with a conservative force can be written Fx=-dU/dx, a plot of U vs. x can tell us something about how a system will behave as a function of position. For relative minima of U vs. x, there will be no force – we call these points stable equilibria. For relative maxima of U vs. x, there will also be no force, but for small displacements away from this point, the force will be away from the equilibrium point – we call these points unstable equilibria.

19. Energy diagrams and stability Example: mass on a spring – stable equilibrium point at x=0

20. Energy diagrams and stability