Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: More gravitational potential energy Conservation of mechanical energy

2. Current homework assignments • WHW7: • In blue Tutorials in Physics homework book • HW-47 #1, HW-48 #2, HW-49 #3, HW-51 #5 • due Wednesday, Oct. 17th in recitation • FHW4: • From end of chapter 7 in University Physics • 7.42, 7.46, 7.66, 7.74 • due Friday, Oct. 26th in recitation

3. Exam 2: Thursday (10/18/07) • Material covered: • Textbook chapters 4, 5, 6, and 7 • Lectures up to and including 10/16 (slides online) • Tutorials on Forces, Newton’s Laws, and Work • Problem Solving Activities 4 - 6 (on Relative Motion, Applying Newton’s Laws) • 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 • Review Session -- SPS -- Tuesday, 5PM, 104 Physics Bldg.

4. Two identical blocks slide down two frictionless ramps. Both blocks start from the same height, but block A is on a steeper incline than block B. The speed of block A at the bottom of its ramp is 1. less than the speed of block B. 2. equal to the speed of block B. 3. greater than the speed of block B. 4. “Can’t tell.”

5. Solution • Which forces do work on block? • Which, if any, are constant? • What is F•Ds for motion?

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

7. 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.

8. 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!

9. 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?

10. 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

11. Hot wheels demo One hot wheels car, car A, rolls down the incline and travels straight ahead, while the other car, B, goes through a loop at the bottom of the incline. When the cars reach the end of their respective tracks, the relative speeds will be: vA > vB vA < vB vA = vB Can’t tell

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

13. 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.

14. 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)

15. 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

16. 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.)

17. A ball of mass m=7 kg attached to a massless string of length R=3 m is released from the position shown in the figure below. (a) Find magnitude of velocity of the ball at the lowest point on its path. (b) Find the tension in the string at that point.

18. A 5.00-kg package slides 1.50 m down a long ramp that is inclined at 12.0 below the horizontal. The coefficient of kinetic friction between the package and the ramp is mk = 0.310. Calculate: (a) the work done on the package by friction; (b) the work done on the package by gravity; (c) the work done on the package by the normal force; (d) the total work done on the package. (e) If the package has a speed of 2.20 m/s at the top of the ramp, what is its speed after sliding 1.50 m down the ramp?

19. 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...

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

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

22. 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.

23. Reading assignment • Elastic energy • Impulse and Momentum • 7.2, 8.1 - 8.2 in textbook