Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Finish gravitational potential energy Springs, elastic energy Conservation of mechanical energy

2. Current homework assignments • WHW7: • In blue Tutorials in Physics homework book • HW-47 #1, HW-48 #2, HW-50 #4, HW-51 #5 • due Wednesday, Oct. 18th in recitation • FHW4: • From end of chapter 7 in University Physics • 7.40, 7.42, 7.66, 7.74 • due Friday, Oct. 27th in recitation

3. Exam 2: Thursday (10/19/06) • Material covered: • Textbook chapters 4, 5, 6, and 7 • Lectures up to and including 10/17 (slides online) • Tutorials on Forces, Newton’s Laws, and Work • Problem Solving Activities 4 - 7 (on Relative Motion, Applying Newton’s Laws, Work) • Homework assignments • As with Exam 1, Exam 2 is closed book, but you may bring calculator and one handwritten 8.5” x 11” sheet of notes -- this may be a different sheet from Exam 1. • Practice versions of Exam 2 posted online

4. Other Announcements • No lecture next Tuesday (10/24/06) -- Eid Ul-Fitr • next lecture = next Thursday (10/26/06) • No recitation workshops next Wednesday (10/25/06) • there is recitation this Friday (the day after Exam 2), then the next one will be next Friday (10/27/06) • No WHW set due next week • WHW8 will be due 11/1/06

5. Work done by gravity Work W = -mg j•Ds Therefore, W = -mgDh N does no work! N Ds mg j i

6. A block is released from rest on a frictionless incline. The block travels to the bottom of the left incline and then moves up the right incline which is steeper than the left side. The maximum height that the block reaches on the right incline is 1. less than 2. equal to 3. greater than the height from which it was released on the left.

7. Solution • Change in kinetic energy on way down depends on initial height (work is path independent) • Equal amount of KE must be lost going up. By W-KE, this means work done by gravity equal and opposite (and path independent). • Therefore same height reached!

8. Stopped-pendulum demo • Pendulum swings to same height on other side of vertical • What if pendulum string is impeded ~1/2-way along its length? Will height on other side of vertical be: • Greater than original height • Same as original height • Less than original height?

9. Curved ramp s = W = F•s = Work done by gravity between 2 fixed pts does not depend on path taken! Work done by gravitational force in moving some object along any path is independent of the path depending only on the change in vertical height

10. Hot wheels demo • Final speed of cars does not depend on shape of track – only net change in vertical height.

11. Work done on an object by gravity W(on object by earth) = – m gDh, where Dh = hfinal – hinitial is the change in height.

12. Defining gravitational potential energy The change in gravitational potential energy of the object-earth system is just another name for the negative value of the work done on an object by the earth.

13. Gravitational Potential Energy • For an object of mass m near the surface of the earth: • Ug = mgh • h is height above arbitrary reference line • Measured in Joules -- J (like kinetic energy)

14. Total energy for object moving under gravity • W-KE theorem now reads: • D(K + Ug) = 0  E = Ug + K = constant • * E is called the (mechanical) energy • * It is conserved: (½) mv2 + mgh = constant

15. Pendulum demo • Energy (K+U) should be constant • If pendulum released with zero speed, will return to same point (height) with zero speed (ignoring air drag, friction, etc.)

16. Conservative forces • If the work done by some force (e.g. gravity) does not depend on path the force is called conservative. • Then gravitational potential energy Ug only depends on (vertical) position of object Ug = Ug(h) • Elastic forces also conservative – elastic potential energy U = (1/2)kx2...

17. Nonconservative forces • friction, air resistance,... • Potential energies can only be defined for conservative forces

18. Conservation of (mechanical) energy • If we are dealing with a potential energy corresponding to a conservative force 0 = DK+DU • Or K + U = constant

19. Conservation of total energy The total energy of an object or system is said to be conserved if the sum of all energies (including those outside of mechanics that have not yet been discussed) never changes. This is believed always to be true.

20. Non-constant force… Work done in small interval Dx DW = F Dx Total W done from A to B  SF Dx = Area under curve! F F(x) x B A Dx

21. Springs -- Elastic potential energy Force F = -kx (Hooke’s law) frictionless table F Area of triangle lying under straight line graph of F vs. x = (1/2)(+/-x)(-/+kx) x F = -kx W = -(1/2)kx2

22. (Horizontal) Spring frictionless table • x = displacement from relaxed state of spring • Work done by spring on block, W = -(1/2)kx2 • Elastic potential energy stored in spring: U = (1/2)kx2 (1/2)kx2 + (1/2)mv2 = constant

23. Motion? • Consider release from rest at x = a – what happens? (1/2)ka2 = (1/2)kx2 + (1/2)mv2 or (1/2)mv2 = (1/2)ka2 - (1/2)kx2 • what range does x lie in? • when is v greatest? • describe motion…

24. Many forces • For a particle which is subject to several (conservative) forces F1, F2 … E = (1/2)mv2 + U1 + U2 +… is constant • Principle called Conservation of total mechanical energy

25. A compressed spring fires a ping pong ball vertically upward. If the spring is compressed by 1 cm initially the ball reaches a height of 2 m above the spring. What height would the ball reach if the spring were compressed by just 0.5 cm? (neglect air resistance) • 2 m • 1 m • 0.5 m • we do not have sufficient information to calculate the new height

26. Many particles • When system consists of many particles it is only the sum of all the particles energies which remains constant

27. Summary • Total (mechanical) energy of an isolated system is constant in time. • Must be no non-conservative forces • Must sum over all conservative forces • Must sum over all particles making up system

28. Reading assignment • Impulse and Momentum • 8.1 - 8.2 in textbook