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Understanding Spring Potential Energy and the Work-Energy Relationship

This lecture covers the concepts of spring potential energy, work, and the transformations between kinetic, potential, and thermal energies. Key topics include the use of energy diagrams, the relationship between force, displacement, and work, as well as the work-kinetic energy theorem. We will also discuss the role of conservative and non-conservative forces in energy conservation. Homework and reading assignments are provided to reinforce these concepts further.

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Understanding Spring Potential Energy and the Work-Energy Relationship

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  1. Lecture 14 • Chapter 10 • Understand spring potential energies & use energy diagrams • Chapter 11 • Understand the relationship between force, displacement and work • Recognize transformations between kinetic, potential, and thermal energies • Define work and use the work-kinetic energy theorem • Use the concept of power (i.e., energy per time) Goals: Assignment: • HW7 due Wednesday, Mar. 10 • For Tuesday: Read through Ch. 12, Sections 1-3, 5 & 6 Do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia”

  2. m Energy for a Hooke’s Law spring • Associate ½ ku2 with the “potential energy” of the spring

  3. m Energy for a Hooke’s Law spring • Ideal Hooke’s Law springs are conservative so the mechanical energy is constant

  4. Emech Emech K K Energy U Energy U y x Energy diagrams • In general: Ball falling Spring/Mass system

  5. U U Equilibrium • Example • Spring: Fx = 0 => dU / dx = 0 for x=xeq The spring is in equilibrium position • In general: dU / dx = 0 for ANY function establishes equilibrium stable equilibrium unstable equilibrium

  6. Comment on Energy Conservation • We have seen that the total kinetic energy of a system undergoing an inelastic collision is not conserved. • Mechanical energy is lost: • Heat (friction) • Deformation (bending of metal) • Mechanical energy is not conserved when non-conservative forces are present ! • Momentum along a specific directionis conserved when there are no external forces acting in this direction. • Conservation of momentum is a more general result than mechanical energy conservation.

  7. Mechanical Energy • Potential Energy (U) • Kinetic Energy (K) • If “conservative” forces (e.g, gravity, spring) then Emech = constant = K + U During  Uspring+K1+K2 = constant = Emech • Mechanical Energy conserved Before During 2 1 After

  8. 1 2 h 3 0 mass: m -x Energy (with spring & gravity) • Emech = constant (only conservative forces) • At 1:y1 = h ; v1y = 0At 2:y2 = 0 ; v2y= ?At 3:y3 = -x ; v3 = 0 • Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 • Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2 • Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0 • Given m, g, h & k, how much does the spring compress? • Em1 = Em3 = mgh = -mgx + ½ kx2 Solve ½ kx2 – mgx +mgh = 0 Given m, g, h & k, how much does the spring compress?

  9. Energy (with spring & gravity) • When is the child’s speed greatest? (A) At y1 (top of jump) (B) Between y1 & y2 (C) At y2 (child first contacts spring) (D) Between y2 & y3 (E) At y3 (maximum spring compression) 1 mass: m 2 h 3 0 -x

  10. Before During 2 1 After Inelastic Processes • If non-conservative” forces (e.g, deformation, friction) then Emech is NOT constant • After  K1+2 < Emech (before) • Accounting for this loss we introduce • Thermal Energy (Eth , new) where Esys = Emech + Eth = K + U + Eth

  11. Energy & Work • Impulse (Force vs time) gives us momentum transfer • Work (Force vs distance) tracks energy transfer • Any process which changes the potential or kinetic energy of a system is said to have done work W on that system DEsys = W W can be positive or negative depending on the direction of energy transfer • Net work reflects changes in the kinetic energy Wnet = DK This is called the “Net” Work-Kinetic Energy Theorem

  12. v Circular Motion • I swing a sling shot over my head. The tension in the rope keeps the shot moving at constant speed in a circle. • How much work is done after the ball makes one full revolution? (A) W > 0 (B) W = 0 (C) W < 0 (D) need more info

  13. Examples of “Net” Work (Wnet) DK = Wnet • Pushing a box on a smooth floor with a constant force; there is an increase in the kinetic energy Examples of No “Net” Work DK = Wnet • Pushing a box on a rough floor at constant speed • Driving at constant speed in a horizontal circle • Holding a book at constant height This last statement reflects what we call the “system” ( Dropping a book is more complicated because it involves changes in U and K, U is transferred to K )

  14. Changes in K with a constant F • If F is constant

  15. Finish Start q = 0° F Net Work: 1-D Example (constant force) • Net Work is F x= 10 x 5 N m = 50 J • 1 Nm ≡ 1 Joule and this is a unit of energy • Work reflects energy transfer • A force F= 10 Npushes a box across a frictionless floor for a distance x= 5 m. x

  16. mks cgs Other BTU = 1054 J calorie = 4.184 J foot-lb = 1.356 J eV = 1.6x10-19 J Dyne-cm (erg) = 10-7 J N-m (Joule) Units: Force x Distance = Work Newton x [M][L] / [T]2 Meter = Joule [L][M][L]2 / [T]2

  17. Net Work: 1-D 2nd Example (constant force) • Net Work is F x= -10 x 5 N m = -50 J • Work reflects energy transfer • A forceF= 10 Nis opposite the motion of a box across a frictionless floor for a distance x = 5 m. Finish Start q = 180° F x

  18. Work in 3D…. • x, y and z with constant F:

  19. Work: “2-D” Example (constant force) • (Net) Work is Fxx= F cos(-45°) x = 50 x 0.71 Nm = 35 J • Work reflects energy transfer • An angled force, F= 10 N,pushes a box across a frictionless floor for a distance x= 5 m and y= 0 m Finish Start F q = -45° Fx x

  20. A q Ay Ax î Scalar Product (or Dot Product) A·B≡ |A| |B| cos(q) • Useful for performing projections. A î= Ax î  î = 1 î j = 0 • Calculation can be made in terms of components. A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz ) Calculation also in terms of magnitudes and relative angles. A B≡ | A | | B | cosq You choose the way that works best for you!

  21. Scalar Product (or Dot Product) Compare: A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz ) with A as force F, B as displacement Dr and apply the Work-Kinetic Energy theorem Notice: F Dr = (Fx )(Dx) + (Fy )(Dz ) + (Fz )(Dz) FxDx +FyDy + FzDz = DK So here F Dr = DK = Wnet More generally a Force acting over a Distance does Work

  22. “Scalar or Dot Product” Definition of Work, The basics Ingredients:Force (F ), displacement ( r) Work, W, of a constant force F acts through a displacement  r: W = F· r(Work is a scalar) F  r  displacement If we know the angle the force makes with the path, the dot product gives usF cos qandDr If the path is curved at each point and

  23. = + atang aradial + = a a a a a a v v v a a a a Remember that a real trajectory implies forces acting on an object • Only tangential forces yield work! • The distance over which FTangis applied: Work path andtime Ftang Fradial F + = = 0 Two possible options: = 0 Change in the magnitude of Change in the direction of = 0

  24. v ExerciseWork in the presence of friction and non-contact forces • 2 • 3 • 4 • 5 • A box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below. • How many forces (including non-contact ones) are doing work on the box ? • Of these which are positive and which are negative? • State the system (here, just the box) • Use a Free Body Diagram • Compare force and path

  25. Lecture 14 Assignment: • HW7 due Wednesday, March 10 • For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6 do not concern yourself with the integration process

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