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Alpha Particle Scattering and angular momentum

  1. Feb 14, 2019 at 3:28 AM #1
    Statement of the problem :

    "Using the definition L = r ##\times## p, prove that the direction of L is constant for an alpha (##\alpha##) particle whose scattering is shown in the diagram below. "

    alpha.png

    Relevant equations :

    We are aware that the scattering takes place via a central force F = F(r) ##\hat r##. Angular momentum L = r ##\times## p and torque ##\tau = r \times F## (all vectors)


    The attempt at a solution

    I can solve the problem, but not the way it asks. The torque ##\tau = r \times F \Rightarrow \tau = 0## since the force is central : F = F(r) ##\hat r##. Using ##\tau = \frac{dL}{dt} = 0##, inplies that the angular momentum vector L is constant.

    [This is not what the question asks for. It asks to show only the direction of L conserved, from the definition of L : L = r ##\times## p].
     
  2. jcsd
  3. Feb 14, 2019 at 3:41 AM #2

    Orodruin

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    If the vector itself is constant, then its direction must be conserved too ...

    Also, where do you think ##\vec \tau = \vec r \times \vec F## comes from? :rolleyes:
     
  4. Feb 14, 2019 at 3:50 AM #3
    Yes, in that sense I have answered the question, but as I said, not in the way they asked for. I suppose they want one to focus on the direction of L and show that it remains the same.
    I don't know. Torque and angular momentum are definitions, defined as the cross product of the radius vector and the force or the linear momentum, respectively.
     
  5. Feb 14, 2019 at 4:11 AM #4

    Orodruin

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    So you need to relate those two, which you are going to do by computing the time derivative of the angular momentum.
     
  6. Feb 14, 2019 at 4:40 AM #5
    All I am aware of is the standard : ##\frac{dL}{dt} = \frac{d}{dt}(r \times p) = r \times \frac{dp}{dt} = r \times F##.

    It is this that I used to answer why the angular momentum L is constant - viz. the thrque is 0 owing to the fact that the force is central and hence L remains the same.

    But the question asks me to show that the direction of L is a constant for a central force like this. (I am aware that I have already shown this in the form of the vector L itself being a constant).

    I wonder if there is another way.

    For instance, a vector L = ##L\hat L##. Can we show that ##\hat L## is a constant from the definition of L ( = ##r \times p##) for a central force (F(r))?
     
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