7. Conservation of Energy

1. 7. Conservation of Energy

2. Recap – Work & Kinetic Energy Work done by a force on an object along a path from A to B Kinetic energy m – mass of object v – velocity of object

3. Recap – Work-Kinetic Energy Theorem The net work, W, done by the net force on an object equals the change, ΔK, in its kinetic energy. Energy is measured in joules (J): J=N m Work can be positive or negative. Kinetic energy is always positive.

4. Conservative Forces Consider the work done by a force in moving an object along a closed path A conservative force is one for which W = 0 along a closed path. And this is independent of the path taken.

5. Example – A Conservative Force If a ski lift moves upwards by a height h then the work done by gravity on you is given by But when the ski lift returns you to your starting point the work done by gravity is +mgh.

6. Nonconservative Forces If, along a closed path, the work done by a force is nonzero, then the force is called nonconservative. Friction is an example of a nonconservative force.

7. Potential Energy

8. Potential Energy Somehow the work done by a conservative force is “stored” since we can recover it as kinetic energy. This “stored work” is called potential energyU. The change ΔUAB in the potential energy associated with a conservative force is the negative of the work done by the force

9. Potential Energy The potential energy difference is defined to be the negative of the work because if the conservative force does positive work then its “store of work”, that is, the potential energy must decrease. Note, according to this definition only differences in potential energy matter.

10. Elastic Potential Energy When a spring is compressed, the work done against the spring force is stored in the spring as potential energy

11. Potential-Energy Function By definition, the potential energy difference is that is, it is the difference between some function U evaluated at points A and B. The function U = U0 + ΔU is called the potential energy function, or just the potential energy.

12. Potential-Energy Functionof Gravity The potential-energy function associated with gravity (taking +y to be up) is given by The value of U0 = U(y0) can be set to any convenient value.

13. Potential-Energy Function of a Spring By convention, one chooses U0 =U(0) = 0

14. Conservation of Energy

15. Conservation of Energy Energy can be neither created nor destroyed Closed System For a closed system the change in energy is zero Open System

16. Conservation of Mechanical Energy According to the work-kinetic energy theorem ΔK = Wnet. The net work can be split into the work Wc done by conservative forces and work Wncdone by nonconservative forces ΔK = Wc + Wnc ΔK = –ΔU + Wnc or ΔK + ΔU = Wnc

17. Conservation of Mechanical Energy ΔK + ΔU = Wnc We see that the change in kinetic energy plus the change in potential energy is equal to the work done by nonconservative forces. If all the forces are conservative, then Wnc = 0. In this case, K – K0 + U – U0 = 0, that is, K + U = K0 + U0

18. Conservation of Mechanical Energy The sum K + U of the kinetic and potential energyis called the mechanical energy. The rule K + U = K0 + U0 expresses the law of conservation of mechanical energy

19. Example (1) How high does the block go? Initial mechanical energy of system Final mechanical energy of system

20. Example (2) Since the forces are conservative the mechanical energy is conserved Height reached

21. Example – Pushing a box up a slope (1) To push a box of mass m a distance d up a slope, the amount of work you do is W. What is the coefficient of friction, assuming you move the box at constant velocity? If F is the force you exert, then the work you do is y x θ

22. Example – Pushing a box up a slope (2) If the force you exert is constant, then we can write y How do we compute F? We use the fact that the box does not accelerate. In this case, x so θ

23. Example – Pushing a box up a slope (3) The work you do is therefore x The magnitude of the frictional force is θ

24. Example – Pushing a box up a slope (4) …putting together the pieces y we get x Note, when θ = 0, we get μk = W/wd, as expected θ

25. Nonconservative Forces

26. Nonconservative Forces The effect of nonconservative forces, such as friction, is to remove energy from a system. Energy, of course, is still conserved if we account for the work done Wnc by these forces: K + U = K0 + U0 + Wnc

27. Example – Sliding a box up a slope (1) How far up a slope does the box go, if its initial velocity is v and the coefficient of friction is μk? Start with K + U = K0 + U0 + Wnc which can be written as ΔK + ΔU = Wnc x θ

28. Example – Sliding a box up a slope (2) ΔK +ΔU = Wnc For this problem, we have x from which we can find d θ

29. Potential-Energy Curves

30. Potential-Energy Curve The potential-energy curve is a plot of the potential energy function U(x) as a function of position x. The figure shows the potential energy curve for a roller-coaster car.

31. Potential-Energy Curve In general, the roller-coaster car has both kinetic K(x) and potential energy U(x). If the mechanical energy is conserved, then the total energy E = K(x) + U(x) = constant Note, to get as far as point D, starting at point A,we need enough energy to get over the potential barrier at C.

32. Force and Potential Energy For small displacements, by definition, ΔUx = –Fx Δx if Fx is a conservative force. This can be re-written as Fx = –ΔU/Δx. Therefore, in the limit Δx→ 0, we obtain the very important result: A conservative force can be written as the derivative of a potential

33. Summary • Potential energy is stored energy that can be converted to kinetic energy. • The change in potential energy is the negative of the work done by a conservative force. • The total mechanical energy K + U is conserved in the absence of nonconservative forces

34. Summary • Potential-energy curves describe potential energy as a function of position or configuration. • A conservative force can be expressed as the negative derivative of potential energy: Fx= –dU/dx