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Chapter 7 Potential energy and conservation of energy

Chapter 7 Potential energy and conservation of energy In this chapter we will explore one of the most fundamental concepts in physics, energy , and one of the most important principles, conservation of energy.

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Chapter 7 Potential energy and conservation of energy

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  1. Chapter 7 Potential energy and conservation of energy In this chapter we will explore one of the most fundamental concepts in physics, energy , and one of the most important principles, conservation of energy. When a conservative force does work, this work is stored in the form of potential energy (symbol: U) and can be retrieved. This is not the case with non-conservative forces.(7-1)

  2. y 1 B A 2 O x Potential Energy U U is defined for conservative forces only Point A (xA , yA) Point B (xB , yB) WAB = work done by a conservative force F from A to B WAB does not depend on the path taken. This can be expressed mathematically as follows: WAB = W(A,B) Also it can be written as: WAB = U(A) - U(B) U is the potential energy. Units: J (7-2)

  3. m F . . . O A B xo x1 path x-axis (7-3) We assume that F depends on position i.e. F = F(x) Point A at xo is called the “reference point” . The choice of xo and U(xo) does not affect the result of our calculations. Thus we will choose xo and U(xo) for maximum convenience

  4. y . path A m dy h mg . O floor Example: Potential energy of the gravitational force Reference point: Point O at yo = 0 Point A at y1 = h We choose: U(y0) = 0 Thus we get : (7-4)

  5. m F(x) . . . O R P xo x path x-axis Summary If we know F(x) along the x-axis we can determine U at any point P using the following equation: The equation above gives U(x) for every point P on the x-axis with coordinate x. F(x)  U(x) (7-5)

  6. g(x) x’ a x O What about the reverse problem? i.e. if we know U(x) can we get the force F(x)? Our starting point is the definition of U: Theorem from calculus: In our case U plays the role of function f and -F plays the role of function g (7-6)

  7. vi vf m F . . . O A B xi xf path x-axis Conservation of mechanical energy (7-7)

  8. vi vf m F . . . O A B xi xf path x-axis (7-8)

  9. Example: An object of mass m is shot straight up with an initial velocity vo . Determine the maximum height h of the object The only force acting on the object is the gravitational force. The gravitational force is conservative. y B . m h mg vo . A floor U = 0 (7-9)

  10. Potential energy of a spring (spring constant k) (7-10)

  11. a “before” b “after” Example (7-3) page 178 In the Atwood machine shown in the figure m1 = 1.37 kg, m2 = 1.51 kg. The system is released from rest with h2 = 0.84 m. Find the speed v of m2 just before it hits the floor (7-11)

  12. U . . A B E x xA xB motion allowed Energy diagram An energy diagram is a plot of the potential energy U(x) versus x The blue horizontal line is the total energy E = U + K (7-12)

  13. . . f C A . B x O xA xB xC (7-13) Maxima and minima of a function f(x) The function f(x) plotted in the figure has one minimum at point B and two maxima at points A and C

  14. U . B . . C A x O xA xB xC Equilibrium Consider an object that moves along the x-axis under the action of a force whose potential energy U is plotted in the figure. (7-14)

  15. U . . C B FC FB . A xB xA xC FA = 0 (7-15) x

  16. . U A . . C B FC FA = 0 FB x xA xB xC (7-16)

  17. Summary 1.Motion is allowed for parts of the x-axis: 2. At turning points (points A and B) U = E At the turning points v = 0 and K = 0 E = U+K 3. At any point on the plot: 4.Minima of the energy diagram correspond to positions of stable equilibrium. Maxima of the energy diagram correspond to positions of unstable equilibrium (7-17)

  18. Motion in two dimensions (7-18)

  19. The reverse problem: If we know U(x,y,z)can we determine F ? The recipe: (7-19)

  20. We will encounter problems that fall in the following two categories: 1. All forces acting on the object(s) under study are conservative. In this case we use the expression: or simply: 2. Some forces acting on the object(s) under study are conservative and some are non-conservative. Fnet = Fc + Fnc Here Fc is the vector sum of all conservative forces, and Fnc is the vector sum of all non-conservative forces. In this case we use the equation: WhereWnc is the work done by Fnc (7-20)

  21. Example (7-6) page 188 A ball of mass m = 10 kg is attached to a wire of length L = 5 m that can swing freely from a support. The ball is pulled aside so that the wire makes an angle 1 = 31  from the vertical. After 10 swings the maximum angle that the ball reaches is 2 = 25  Calculate the work Wf done by air resistance (a non-conservative force) on the ball during these 10 swings (7-21)

  22. Calculation of the potential energy U for the ball U = mgh h = OA - OB OA = L O C B m h A From triangle OBC we have; OB = OCcos OC = L Thus OB = Lcos and h = L(1 - cos)  U = mgL(1 - cos) (7-22)

  23. 1. “Before” 2. “After” 1 2 E = Wf E = E2 - E1 E1 = mgL(1 - cos1) E2 = mgL(1 - cos2)  E = mgL(cos1 - cos2) Wf = mgL(cos1 - cos2) = 109.85 [cos(31) -cos(25)] Wf = -24 J (7-23)

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