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Understanding Spring Potential Energies and Energy Diagrams

Learn about Hooke's Law spring energy, equilibrium positions, and energy conservation in collisions. Explore the relationship between forces, displacements, work, and transformations of mechanical energies. Understand the concept of mechanical energy and conservation laws. Solve energy-related problems and analyze work-energy theorems.

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Understanding Spring Potential Energies and Energy Diagrams

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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: • HW6 due Wednesday, Mar. 11 • For Tuesday: Read Chapter 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 kx2 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 K Energy U y Energy diagrams • In general: Ball falling Spring/Mass system Emech K Energy U x

  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) • Bending of metal and deformation • Kinetic energy is not conserved by these non-conservative forces occurring during the collision ! • Momentum along a specific directionis conserved when there are no external forces acting in this direction. • In general, easier to satisfy conservation of momentum than 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. 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 1 2 h 3 0 mass: m -x

  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. Energy (with spring & gravity) • When is the child’s speed greatest? • A: Calculus soln. Find v vs. spring displacement then maximize (i.e., take derivative and then set to zero) • B: Physics: As long as Fgravity > Fspring then speed is increasing Find whereFgravity- Fspring= 0 -mg = kxVmaxor xVmax = -mg / k So mgh = Ug23 + Us23 + K23 = mg (-mg/k) + ½ k(-mg/k)2 + ½ mv2  2gh = 2(-mg2/k) + mg2/k+ v2 2gh + mg2/k = vmax2 1 2 h 3 kx mg 0 -x

  11. 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

  12. 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

  13. 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

  14. 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 )

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

  16. 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

  17. 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

  18. 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

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

  20. 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 • A force F= 10 Npushes a box across a frictionless floor for a distance x= 5 m and y= 0 m Finish Start F q = -45° Fx x

  21. 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!

  22. 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

  23. “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

  24. = + 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

  25. 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? • Use a Free Body Diagram • Compare force and path

  26. Lecture 14 Assignment: • HW6 due Wednesday, March 11 • For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6 do not concern yourself with the integration process

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