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Energy conservation

  1. Apr 24, 2009 #1
    1. I have to use energy conservation to determine the forces acting on the particles.


    2. An isolated system consists of two particles of masses m1 and m2, whose position vectors in an inertial frame are x1 and x2 and velocity vectors are v1 and v2.

    The interaction of the particles can be described by the energy function :

    E= 1/2m1(v1)^2 + 1/2m2(v2)^2 - (k/r^2)

    k is a positive constant
    r = mod(x1-x2) and is the magnitude of the seperation vector.

    3. Do I have to differentiate E to get 1/2m1(a1)^2 + 1/2m2(a2)^2 - (k/r^2) and equate it to zero?
     
    Last edited: Apr 24, 2009
  2. jcsd
  3. Apr 24, 2009 #2
    The units of the first two expressions involving the mass are not energy but momentum. Check the expression again.
     
  4. Apr 24, 2009 #3
    Ooops. Thanks chrisk for pointing that out.


    Indeed the velocity vectors should each be squared.
     
  5. Apr 24, 2009 #4
    Differentiating with respect to time does lead to

    dE/dt = 0

    because the total energy of the system is constant. Take into account that r is a function of time.
     
  6. Apr 24, 2009 #5
    So dE/dt = m1a1 + m2a2 +2k/mod(v1-v2)^3 ?

    Would I be right in saying that due to superposition principle then the forces on the particles is F = -2k/mod(v1-v2)^3 ?
     
  7. Apr 24, 2009 #6
    Check how you differentiated. Use the chain rule.
     
  8. Apr 24, 2009 #7
    Ah.

    So dE/dt = m1a1 +m2a2 + [2k/mod(x1-x2)^3](v1 -v2) = 0
     
  9. Apr 24, 2009 #8
    Recall that

    F=-dU/dx

    when F is conservative. The given expression contains kinetic and potential energy (U) terms.
     
  10. Apr 24, 2009 #9
    Then F= [-2k/mod(x1-x2)^3](v1-v2)
     
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