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Torque and angular momentum

  1. Mar 26, 2016 #1
    1. The problem statement, all variables and given/known data
    The torque ## \vec{τ} ## on a body about a given point is found to be equal to ## \vec{A} × \vec{L} ## where ##\vec{A}## is a constant vector, and ##\vec{L}## is the angular momentum of the body about that point. From this it follows: (Multiple answers correct)

    (A) ## \dfrac{d\vec{L}}{dt} ## is perpendicular to ##\vec{L}## at all instants of time

    (B) the component of ##\vec{L}## in the direction of ##\vec{A}## does not change with time

    (C) the magnitude of ##\vec{L}## does not change with time

    (D) ##\vec{L}## does not change with time

    2. Relevant equations
    ##\vec{τ} = \dfrac{d\vec{L}}{dt} ##

    3. The attempt at a solution
    ##\vec{τ} = \dfrac{d\vec{L}}{dt} = \vec{A} × \vec{L} ##
    From this equation (A) holds.

    (D) will hold, i.e. only if ##\dfrac{d\vec{L}}{dt}## is 0, i.e.## \vec{A} ## is parallel to ## \vec{L} ## which has no reason to be true all the time. So, D should not be correct.

    Which leaves (B) and (C). I have no idea how to prove or disprove them. Please help.
     
  2. jcsd
  3. Mar 26, 2016 #2

    Orodruin

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    Why don't you simply try finding some expressions for the vectors relevant to B and C and differentiate them with respect to time?
     
  4. Mar 26, 2016 #3
    Thank you! Got it. I differentiated these two expressions:
    For (B)
    ##\vec{L}.\vec{L} = L^2##
    and for (C)
    ##\vec{L}.\vec{A} / A ##
     
  5. Mar 26, 2016 #4
    Hence, (A) (B) and (C) are true.
     
  6. Mar 26, 2016 #5

    Orodruin

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    Correct.
     
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