1. Solution: Physics 1710 Chapter 8—Potential Energy Power = dW/dt = (Fdx)/dt = F dx/dt = F v (for F constant) = (20.0 N )(36 x 10 3 m/ 3600 sec) = 200. N m/s = 200. W (#4)

2. 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? R h Physics 1710 Chapter 8—Potential Energy Review

3. 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? R h v 2 /R = g K = U - W ½ mv 2 + ½ ( 2/5 mv 2 )= mgh v 2 = Rg = 10/7 hg h = 0.7 R h = 0.7 R h = 7/10 (22.0 cm) = 15.4 cm d = 2R+h = 69.4 cm v Physics 1710 Chapter 8—Potential Energy d Physics Works! (When you include all relevant effects)

4. 1′ Lecture Potential Energy is U = -∫ Fd r Potential Energy is U = -∫ Fd r The sum of all energy, potential and kinetic, of The sum of all energy, potential and kinetic, of a system is conserved, in the absence of dissipation: a system is conserved, in the absence of dissipation: E = U + K – W F = - ∇U = negative gradient of U. F = - ∇U = negative gradient of U. Physics 1710 Chapter 8—Potential Energy

5. Potential Energy: W = ∫ Fd r U = -W = -∫ Fd r Potential Energy is the negative of the work required to put the system in the current state. Potential Energy is the negative of the work required to put the system in the current state. Physics 1710 Chapter 8—Potential Energy

6. What is the potential energy of a 0.100 kg ball placed up a 45 o ramp 0.50 above the table? Physics 1710 Chapter 8—Potential Energy F x 0.50 m

7. What is the potential energy of a 0.100 kg ball placed up a 45 o ramp 0.50 above the table? Physics 1710 Chapter 8—Potential Energy F x 0.50 m - mg U = - F‧x = - (-mg)h = mg h

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

9. Relationship Between F and U: U = -∫ Fd r So U = -∫ [ F x dx + F y dy + F z dz] Then F x =-dU/dx ; F y =-dU/dy; F z =-dU/dz F = -∇U F= -gradient of U Physics 1710 Chapter 8—Potential Energy

10. The Force is equal to the negative gradient of the potential energy: F = -∇U F x = -∂U/∂x F y = -∂U/∂y F y = -∂U/∂y F z = -∂U/∂z Physics 1710 Chapter 8—Potential Energy

11. Example: Pendulum U = mg h h = L(1- cos  ) U = mg L(1- cos  ) s= L  s= L  F S = - (1/L)dU/d  = - mg sin  = - mg sin  Physics 1710 Chapter 8—Potential Energy s L 

12. Example:Ball on a slope h = ax + by h = ax + by U = mgh U = mgh F x = -∂U/∂x = -∂(mgh)/∂x = -mg∂h/∂x F x = -∂U/∂x = -∂(mgh)/∂x = -mg∂h/∂xSimilarly: F y = -∂U/∂y = -mg b F y = -∂U/∂y = -mg b Thus, F = -mg( a i + b j ) Thus, F = -mg( a i + b j ) Physics 1710 Chapter 8—Potential Energy

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

14. Force: z = ar 2 z = ar 2 U = mgz U = mgz F r = -∂U/∂r = -∂(mgz)/∂r = -mg∂z/∂r F r = -∂U/∂r = -∂(mgz)/∂r = -mg∂z/∂r = - 2amgr = - k r = - 2amgr = - k r Like a mass on a spring! Physics 1710 Chapter 8—Potential Energy

15. Conservation of Energ y: The sum of all energy in a system is conserved, i.e. remains the same. The sum of all energy in a system is conserved, i.e. remains the same. E = U + K Physics 1710 Chapter 8—Potential Energy

16. Dissipative (non-conservative) Forces: W = ∫ Fd r =∫ (C v x 2 )dx =∫ (C v x 2 )(dx /dt) dt =∫ (C v x 3 )dt E = U + K -W Physics 1710 Chapter 8—Potential Energy

17. Summary: The Potential Energy is equal to the negative of the work done on the system to put it in its present state.The Potential Energy is equal to the negative of the work done on the system to put it in its present state. U = -∫ Fd r F = - ∇U F = - ∇U The sum of all energy, potential and kinetic, of a system is conserved, in the absence of dissipation. The sum of all energy, potential and kinetic, of a system is conserved, in the absence of dissipation. E = U + K – W Physics 1710 Chapter 8—Potential Energy

18. Potential Energy: F -F h 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 Physics 1710 Chapter 8—Potential Energy