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Understanding Spring Potential Energies in Physics

Learn about Hooke's Law springs, energy diagrams, equilibrium positions, and conservation of mechanical energy in physics. Explore energy transformations, work-kinetic energy theorem, and power concepts.

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Understanding Spring Potential Energies in Physics

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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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