Conservation of Energy and Conservative Forces

1. Thursdays 12 – 2 pm PPP “Extension” lecture. Room 211 podium level Turn up any time 7.1-7.6 Work and Kinetic energy 8.2 Potential energy 8.3 Conservative Forces and Potential energy 8.5 Conservation of Mech. Energy 8.6 Potential-energy curves 8.8 Conservation of Energy Systems of Particles 9.2 Centre of mass Summary Lecture 7 Problems: Chap. 8 5, 8, 22, 29, 36, 71,51 Chap. 9 1, 6, 82

2. Outline Lecture 7 Work and Kinetic energy Work done by a net force results in kinetic energy Some examples: gravity, spring, friction Potential energy Work done by some (conservative) forces can be retrieved. This leads to the principle that energy is conserved Conservation of Energy Potential-energy curves The dependence of the conservative force on position is related to the position dependence of the PE F(x) = -d(U)/dx

3. Kinetic Energy Work-Kinetic Energy Theorem Changein KE work done byallforces DK  Dw

4. SF Vector sum of all forces acting on the body xf xi x Work-Kinetic Energy Theorem = 1/2mvf2 – 1/2mvi2 = Kf - Ki = DK Work done by net force = change in KE

5. h F mg Gravitation and work Work done by me (take down as +ve) = F.(-h) = -mg(-h) = mgh Work done by gravity = mg.(-h) = -mgh ________ Total work by ALL forces (W) = 0 =DK Lift mass m with constant velocity Work done by ALL forces = change in KE DW = DK What happens if I let go?

6. F -kx x Compressing a spring Compress a spring by an amount x Work done by meFdx = kxdx = 1/2kx2 Work done by spring-kxdx =-1/2kx2 0 Total work done (DW)= =DK What happens if I let go?

7. F f d Moving a block against friction at constant velocity Work done by me = F.d Work done by friction = -f.d = -F.d Total work done = 0 What happens if I let go? NOTHING!! Gravity and spring forces are Conservative Friction is NOT!!

8. Conservative Forces A force is conservativeif the work it does on a particle that moves through a round trip is zero: otherwise the force is non-conservative A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points: otherwise the force is non-conservative.

9. Conservative Forces h -g Sometimes written as A force is conservativeif the work it does on a particle that moves through a round trip is zero; otherwise the force is non-conservative Consider throwing a mass up a height h work done by gravity for round trip: On way up: work done by gravity = -mgh On way down: work done by gravity = mgh Total work done = 0

10. Conservative Forces h -g A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points: otherwise the force is non-conservative. Work done by gravity w = -mgDh1+ -mgDh2+-mgDh3+… Each step height=Dh = -mg(Dh1+Dh2+Dh3 +……) = -mgh Same as direct path (-mgh)

11. Conservation of Energy

12. Potential Energy h mg The change in potential energy is equal to minus the work done BY the conservative force ON the body. DU = -Dw Work done by gravity = mg.(-h) = -mgh Therefore change in PE is DU = -Dw Ugrav = +mgh Lift mass m with constant velocity

13. Potential Energy F -kx x The change in potential energy is equal to minus the work done BY the conservative force ON the body. Compress a spring by an amount x Work done by spring is Dw = -kx dx = - ½ kx2 Therefore the change in PE is DU = - Dw Uspring = + ½ kx2

14. Potential Energy decrease n increase The change in potential energy is equal to minus the work done BY the conservative force ON the body. DU = -Dw but recall that Dw = DK so that DU = -DK or DU + DK = 0 Any increase in PE results from a decrease in KE

15. DU + DK = 0 In a system of conservative forces, any change in Potential energy is compensated for by an inverse change in Kinetic energy U + K = E In a system of conservative forces, the mechanical energy remains constant

16. = F. x In the limit thus Potential-energy diagrams Dw= - DU The force is the negative gradient of the PE curve If we know how the PE varies with position, we can find the conservative force as a function of position

17. Energy U= ½ kx2 F = -kx (spring force) x PE of a spring here U = ½ kx2

18. Energy Total mech. energy KE U= ½ kA2 E= ½ kA2 x PE x’ x=A Potential energy U= ½ kx2 At any position x PE + KE = E U + K = E K = E - U = ½ kA2 – ½ kx2 = ½ k(A2 -x2)

19. K Et U Roller Coaster Fnet = mg – R R = mg - Fnet Fnet=-dU/dt

20. K Et R U mg Fnet = mg – R R = mg - Fnet Fnet=-dU/dx

21. Conservation of Energy We said:when conservative forces act on a body DU + DK = 0 U + K = E(const) This would mean that a pendulum would swing for ever. In the real world this does not happen.

22. Energy converted to other forms Conservation of Energy When non-conservative forces are involved, energy can appear in forms other than PE and KE (e.g. heat from friction) DU + DK + DUint = 0 Ki + Ui = Kf + Uf + Uint Energy may be transformed from one kind to another in an isolated system, but it cannot be created or destroyed. The total energy of the system always remains constant.

23. mg f h v0 upward Stone thrown into air, with air resistance. How high does it go? Ei = Ef + Eloss Ki + Ui = Kf + Uf + Eloss ½mvo2 + 0 = 0 + mgh + fh ½mvo2 = h(mg + f)

24. f mg mg = ½mvf2 + f h ½mvf2 = mg - f downward Stone thrown into air, with air resistance. What is the final velocity ? E’i = E’f + E’loss K’i + U’i = K’f + U’f + E’loss 0 + mgh = ½mvf2 + 0 + fh

25. System of Particles

26. That's all folks

27. M m1 m2 0 x1 x2 xcm Centre of Mass (1D) M = m1 + m2 M xcm = m1 x1 + m2 x2 In general