8-1 Conservative and Nonconservative Forces Definition 1: The total work around a closed path is zero for a conservative force. Work done by gravity = -mghWork done by gravity = +mgh Total work around a closed path = -mgh + mgh = 0. The force of gravity is a conservative force. Ch 8 Potential Energy and Conservation of Energy

Conservative Force Definition 2: The work is independent of the path for a conservative force. W 1 + W 2 = 0 W 1 + W 3 = 0 W 2 = W 3 = -W 1

WAWA WBWB Path 1 W Total WAWA WBWB Path 2 W Total P8.3a (p.233) F F x x A A B B

Concept Question 1: P8.3 If the mass increases, the work done by the spring will A. increase. B. decrease. C. stay the same.

8-2 Potential Energy and the Work Done by Conservative Forces A. Potential Energy W C = -  U W C = the work of the conservative force B. Gravity U = mgy with U = 0 at y = 0 near the Earth’s surface P8.57 (p.237) C. Springs (ideal) U = kx 2 /2 with U = 0 at x = 0. P8.9 (p.234)

Concept Question 2: P8.57 The change in U g from 90 to 45 degrees is A. greater than that from 45 to 0 degrees. B. less than that from 45 to 0 degrees. C. the same as that from 45 to 0 degrees.

8-3 Conservation of Mechanical Energy Mechanical Energy E = U + K W =  K (Ch. 7) For a conservative force W C = -  U -  U =  K  U +  K = 0 =  E E is constant or conserved P8.17 (p.234)

Concept Question: Consider a system where the force is all conservative. The initial kinetic energy is 4 J, the final kinetic energy is 8 J and the final potential energy is 5 J. What was the initial potential energy? A. 8 J B. 9 J C. 9 N D. 8 N E. 5 J

Concept Question 5: P8.17 If the mass doubles, the change in height of the ball A. double. B. halve. C. stay the same.

P8.22 (p.235) Similar to P8.18

8-4 Work Done by Nonconservative Forces W TOT = W C + W NC =  K (Ch. 7) -  U + W NC =  K W NC =  K +  U W NC =  E

P8.86 (p.239)

Concept Question 4: P8.86 If the mass is multiplied by 4, the speed will A. quadruple (4x) B. double. C. stay the same. D. halve. E. quarter (1/4).

A Ball Rolling on a Frictionless Track 8-5 Potential Energy Curves and Equipotentials

Gravitational Potential Energy Versus Position for the Track Shown in the Previous Slide

A Mass on a Spring U = kx 2 /2

Conceptual Questions: An object starts from point A. CT6: The speed at A is the same as at B,C,D,E,F,G? CT7: The speed is the greatest at B,C,D,E,F,G?

A Contour Map – an example of an equipotential plot All the numbered lines are at the same gravitational potential energy.

P8.46 (p.236) 8.6J 4.7m 0.3m

P8.81 (p.238)

P8.83 (p.239)