1. Mechanics 105  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 Potential energy (chapter seven)

2. Mechanics 105 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 y a to y b is:

3. Mechanics 105 Potential energy This work is a transfer of energy to the system. We can define the quantity U g  mgy to be the gravitational potential energy The work done on the system then gives the change in U g : W=  U g change 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. Mechanics 105 Isolated systems Consider the system of an object only in the earth’s gravitational field, falling from y b to y a. In free fall, the work done by gravity is mg(y b -y a ), 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. Mechanics 105 Isolated systems  We can write this as a continuity equation for the mechanical energy E mech =K+Ug  E mech =0, or K i +U i =K f +U f

6. Mechanics 105 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 U g =0)  At height y, what is the speed?

7. Mechanics 105 Example – object in free fall Initial energy = K+U g =mgh Final energy (at any point y) = mgy+½mv 2 mgh=mgy+½mv 2  v=(2g(h-y)) ½ y U g =mgy K=½mv 2 y=h U g =mgh K=0 y=0 U g =0

8. Mechanics 105  Conceptest  Demo

9. Mechanics 105 Example If m 1 >m 2, How fast will m 1 be going when it hits the floor? Start: K+U g =m 1 gh End: ½ m 1 v 2 +m 2 gh m 1 gh= ½ m 1 v 2 +m 2 gh v=[2gh(m 1 -m 2 )/m 1 ] ½ m1m1 m2m2

10. Mechanics 105 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. Mechanics 105 Conservative and nonconservative forces Potential energy of a spring U s  ½ kx 2 Gravitational potential energy U g  mgh New formulation of Work-KE thm: K+U+E int =constant Conservation of energy

12. Mechanics 105 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? K i +U i = K f +U f 0+mgh= ½ mv 2 +0  v=(2gh) ½

13. Mechanics 105 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 ·d r From this we can see that a conservative force can be written as F x =-dU/dx ( F =-  U) e.g. F g =-dU g /dy=-d(mgy)/dy=-mg

14. Mechanics 105 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. Mechanics 105 Potential energy for gravitational and electrostatic forces Gravitational force between two masses (m 1, m 2 ) separated by a distance r This gives a general form for the gravitational potential energy of: And for the electrostatic potential energy

16. Mechanics 105 Example

17. Mechanics 105 Example

18. Mechanics 105 Energy diagrams and stability Since the potential energy associated with a conservative force can be written F x =-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. Mechanics 105 Energy diagrams and stability Example: mass on a spring – stable equilibrium point at x=0

20. Mechanics 105 Energy diagrams and stability

21. Mechanics 105