Physics 1710 Chapter 7&8—Power & Energy

1. 0 Physics 1710 Chapter 7&8—Power & Energy K = ½ mv 2 = ½ (2.0 kg) (5.0 m/s) 2 = 25. kg m2/s2 = 25. J Solution:

2. R 0 Physics 1710 Chapter 7&8—Power & Energy h What is the minimum height from which a small rolling ball must be started from rest so that it will complete a loop-the-loop?

3. No Talking! Think! Confer! 0 Physics 1710 Chapter 7&8—Power & Energy What is the minimum height from which a small rolling ball must be started from rest so that it will complete a loop-the-loop? Peer Instruction Time

4. v R 0 Physics 1710 Chapter 7&8—Power & Energy v2/R = g K = U ½ mv 2 = mg(h-R) v 2 = 2 g (h-R) g R = 2g (h-R) h = 3R h What is the minimum height from which a small rolling ball must be started from rest so that it will complete a loop-the-loop?

5. 0 Physics 1710 Chapter 7&8—Power & Energy What is the minimum height from which a small rolling ball must be started from rest so that it will complete a loop-the-loop? K = ½ mv 2 = ½ (2.0 kg) (5.0 m/s) 2 = 25. kg m2/s2 = 25. J

6. 0 Physics 1710 Chapter 7&8—Power & Energy 1′ Lecture Power is the time rate of change in energy. [Power]= [Watt] = [Joule]/[second] Potential Energy U is the energy stored in a system and may later produce work. The Potential Energy is equal to the negative of the work done on the system to put it in its present state. The sum of all energy, potential and kinetic, is conserved in an isolated system.

7. 0 Physics 1710 Chapter 7&8—Power & Energy Power: P = dE/dt Power is the time rate of change in the energy of a system, the rate of work down on or by the system. Unit of power = Joule/second = Watt

8. 0 Physics 1710 Chapter 7&8—Power & Energy Unit of Work and Energy: [F ‧ d ] = N‧m = Joule = J Joule

9. James Watt 0 Physics 1710 Chapter 7&8—Power & Energy Power: Watt = Joule/second Watt’s Steam Engine 1774

10. 0 Physics 1710 Chapter 7&8—Power & Energy Power: P = dE/dt P = ∆E/∆t ∆E = P∆t ∆E =(100 W)(3600 s) ∆E =360 000 J = 360 kJ

11. 0 Physics 1710 Chapter 7&8—Power & Energy Potential Energy: W = ∫ F•d r U = -W Potential Energy is the negative of the work required to put the system in the current state.

12. -F h F 0 Physics 1710 Chapter 7&8—Power & Energy Potential Energy: U = - (- F h) = m g h

13. 0 Physics 1710 Chapter 7&8—Power & Energy Example: Elevated Mass F = -mg Potential Energy: Ug = -∫0hFdy = -∫0h(- mg) dy Ug = mg∫0h dy = mgh Thus, the potential energy stored in an elevated mass is proportional to the height h and the weight of the mass.

14. No Talking! Think! Confer! 0 Physics 1710 Chapter 7&8—Power & Energy Where does the energy come from to produce electrical power in a hydroelectric dam? Peer Instruction Time

15. -F h F 0 Physics 1710 Chapter 7&8—Power & Energy U = m g h P = dU/dt = mg dh/dt mg =(100. kg)(9.8N/kg) = 98.0 N dh/dt = 10 m/10 s = 1 m/s P = 98. W Potential Energy:

16. 0 Physics 1710 Chapter 7&8—Power & Energy Relationship Between F and U: U = -∫ F•d r So U = -∫ [ Fx dx + Fy dy + Fz dz] Then Fx =-dU/dx ; Fy =-dU/dy; Fz =-dU/dz F = -∇U F= -gradient of U

17. 0 Physics 1710 Chapter 7&8—Power & Energy Example: Mass on a Spring Potential Energy: U = ½ k x 2 F =dU/dx F= -½ k dx2/dx F= -k x Thus, the force is equal to the negative of the gradient of the potential energy.

18. 0 Physics 1710 Chapter 7&8—Power & Energy The Force is equal to the negative gradient of the potential energy: F = -∇U Fx = -∂U/∂x Fy = -∂U/∂y Fz = -∂U/∂z

19. 0 Physics 1710Chapter 8 Potential Energy and Conservation Example: Ball on a slope h = ax + by U = mgh Fx = -∂U/∂x = -∂(mgh)/∂x = -mg∂h/∂x Similarly: Fy = -∂U/∂y = -mg b Thus, F = -mg( ai + b j )

20. 0 Physics 1710 Chapter 7&8—Power & Energy Conservation of Energy: The sum of all energy in a system is conserved, i.e. remains the same. E = U + K

21. 0 Physics 1710 Chapter 7&8—Power & Energy Example: Pendulum U = mg h h = L(1- cos  ) U = mg L(1- cos  ) K = ½ m v 2 =½ m (Ld /dt) 2 E = mg L(1- cos  ) + ½ m (Ld /dt) 2 = constant

22. 0 Physics 1710 Chapter 7&8—Power & Energy Thought (Gedanken) Experiment: Why does a pendulum stop moving?

23. 0 Physics 1710 Chapter 7&8—Power & Energy Dissipative (non-conservative) Forces: W = ∫ F•d r =∫ (C vx2 )dx =∫ (C vx2 )(dx /dt) dt =∫ (C vx3 )dt E = U + K -W

24. 0 Physics 1710 Chapter 7&8—Power & Energy Summary: The Potential Energy is equal to the negative of the work done on the system to put it in its present state. U = -∫ F•d r The sum of all energy, potential and kinetic, of a system is conserved, in the absence of dissipation. E = U + K – W F = - ∇U P = dE/dt