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Energy Methods: Sect. 2.6

Energy Methods: Sect. 2.6. Consider a point particle under the influence of a conservative force in 1d (for simplicity). Conservation of mechanical energy:  E = T+U = ( ½) mv 2 + U(x) = constant Rewrite this, using v = (dx/dt) as:

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Energy Methods: Sect. 2.6

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  1. Energy Methods: Sect. 2.6 • Consider a point particle under the influence of a conservative force in 1d(for simplicity). Conservation of mechanical energy:  E = T+U = (½)mv2 + U(x) = constant • Rewrite this, using v = (dx/dt) as: v(t) = (dx/dt) = [2(E - U(x))/m]½ v(x) Clearly, this requires (for real v): E  U(x) • For general U(x), & given initial position xoat t = to, formally solve the problem by integrating (get t(x), rather than x(t)): (limits: xo x) t - to = ∫dx[2(E - U(x))/m]-½ v as a function of x! 

  2. t as a function of x!  t(x) - to = ∫dx[2(E - U(x))/m]-½ • Given U(x), (in principle)integrate this to get t(x) & (in principle) algebraically invert t(x) to get x(t), which is what we want! • In future chapters, we will do (in detail): 1. The harmonic oscillator: U(x) = (½)kx2 2. Gravitation: U(x) = -(k/x)

  3. General U(x): Learn a lot about particle motion by analyzing plot of U(x) vs. x for different energies E: T = (½)mv20  Any real physical situation requires E = T + U(x)  U(x)

  4. T = (½)mv2  0  E = (½)mv2 + U(x)  U(x) • Consider motion for differentE. E = E1: Bounded & periodic betweenturning points xa & xb. Bounded Particle never gets out of the region xa  x  xb. Periodic:Moving to left, will stop at xa, turn around & move to right until stops at xb, & turn around again, repeating forever. xa & xb are called Turning Points for obvious reasons. Turning Points:v = 0, T= 0, E1 = U(x).Gives xa & xb E = T + U = constant, but T & U change throughout motion.

  5. T = (½)mv2  0  E = (½)mv2 + U(x)  U(x) • Consider motion for different E. E = E2: Bounded & periodic between turning points xc & xd and separately between turning points xe & xf . Bounded The particle never gets out of region xc  x  xd or out of region xe  x  xf. Periodic:Goes from one turning point to another, turns around & moves until stops at another turning point. Repeats forever. Particle is in one valley or another. Can’t jump from one to another without getting extra energy > E2(but, in QM: Tunneling!)Turning points:v = 0, T= 0, E2 = U(x).Gives xc , xd , xe & xf

  6. T = (½)mv2  0  E = (½)mv2 + U(x)  U(x) • Motion for different E: E = E0:Since E = U(x), T= 0 & v = 0. Particle doesn’t move. Stays at x0 forever. x0 is determined by E0 = U(x0) E = E3:If the particle is initially moving to the left, it comes in from infinity to turning pointxg, stops, turns around, & goes back to infinity. Turning pointxgdetermined byv = 0, T=0, E3 =U(x). E = E4:Unbounded motion. The particle can be at any position. Its speed changes as E - U(x) = T = (½)mv2changes.

  7. Motion of particle at energy E1: This is similar to the mass-spring system. • Approximate potential for xa  x  xb is a parabola: U(x)  (½)k(x-x0)2 x0 = equilibrium point (Ch. 3!)

  8. The Approximate Potential for xa  x  xb is a parabola: U(x)  (½)k(x-x0)2 where x0is the Stable Equilibrium Point • For the motion of a particle at energy E: If there are 2 turning points, xa& xb, the situation looks like the figure. xb xa x0

  9. Equilibrium Points • Equilibrium Point  Point where the particle will stay & remain motionless. • Stable Equilibrium PointIf the particle is displaced slightly away from that point, it will tend to return to it. (Like the bottom of parabolic potential well). • Unstable Equilibrium Point If the particle is displaced slightly away from that point, it will tend to move even further away from it. (Like the top of an upside down parabolic barrier.) • Neutral Equilibrium Point If the particle is displaced slightly away from that point, it will tend to stay at new point.(Like a flat potential).

  10. Assume that the equilibrium point is at x = 0. In general, expand U(x) in a Taylor’s series about the equilibrium point [(dU/dx)0  (dU/dx)x0] U(x)  U0 + x(dU/dx)0 + (x2/2!)(d2U/dx2)0 + (x3/3!)(d3U/dx3)0 + ... • By definition, if x = 0 is an equilibrium point, the force = 0 at that point:  F0  - (dU/dx)0 = 0 We can choose U0 = 0 since the zero of the potential is arbitrary. So:U(x)  (x2/2!)(d2U/dx2)0 + (x3/3!)(d3U/dx3)0 + ...

  11. For a general potential U(x), not far from an equilibrium point, keep the lowest order term only: U(x)  (x2/2!)(d2U/dx2)0 Or: U(x)  (½)kx2 where k  (d2U/dx2)0 • Equilibrium conditions: 1. k = (d2U/dx2)0 > 0 :Stable equilibrium. U(x)  Simple harmonic oscillator potential 2. k = (d2U/dx2)0 <0: Unstable equilibrium. 3. k = (d2U/dx2)0 = 0: May be neutral equilibrium, but must look at higher order terms.

  12. Example 2.12 A string, length b, attached at A, passes over a pulley at B, 2d away, & attaches to mass m1. Another pulley, with mass m2 attached passes over string, pulls it down between A & B. Calculate the distance x1when system is in equilibrium. Is the equilibrium stable or unstable? Work on the board!

  13. Example 2.13 Potential:U(x) = -Wd2(x2+d2)/(x4+8d4) Sketch this potential & discuss motion at various x. Is it bounded or unbounded? Where are the equilibrium positions? Are these stable or unstable? Find the turning points for E = -W/8.

  14. Limitations of Newtonian MechanicsSect. 2.7 • Implied assumptions of Newtonian Mechanics: • r, v, t, p, E are all measurable (simultaneously!) • All can be specified with desired accuracy, depending only on the sophistication of our measuring instruments. True for MACROSCOPICobjects! Not true for MICROSCOPIC(atomic & smaller) objects! • Quantum mechanics is needed for these! Heisenberg uncertainty, for example tells us that ΔxΔp  (½)ħ  We cannot precisely know the x & p for a particle simultaneously! • Quantum mechanics   Newtonian mechanics as size of the object increases.

  15. Newtonian mechanics also breaks down when the speed v of a particle approaches a significant fraction of the speed of light c. • Need Special Relativity for these cases (Ch. 14) • The is no concept of absolute time. • Simultaneous events depend on the reference frame. • There is time dilation. • There is length contraction. • Light speed c is limitation on speed of objects.

  16. Practical limitation to Newtonian mechanics: • It is impractical when dealing with systems of huge numbers of particles  1023. • Even with the most sophisticated & powerful computers, we cannot simultaneously solve this many coupled differential equations! • For such problems we need the methods of Statistical Mechanics (Physics 4302). • This uses the methods of probability & statistics to compute average properties of the system.

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