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Circular motion in a magnetic field

  1. Aug 3, 2015 #1
    1. The problem statement, all variables and given/known data
    Small mass ##m## ball has a negative charge ##q## and is hanging on an inelastic string which has a length of ##l##. What is the smallest velocity that one need to impart on this ball for it to make one revolution? There is also a uniform magnetic field ##B## as shown in the drawing.
    Circular_motion.png

    2. Relevant equations
    ##F=Bqv##
    ##F_c=\frac{mv^2}{r}##

    3. The attempt at a solution
    We need to find ##v_o##

    Conservation of energy:
    (1) ##\frac{mv_o^2}{2}=2mgl+\frac{mv^2}{2}##

    The force ##F=Bqv## always points into the center of the circle. When the ball reaches the top of the circle, it will be affected by two forces ##F=Bqv## and ##mg##. Both point downwards, hence the sum of those forces must be the centripetal force.
    (2) ##Bqv+mg=\frac{mv^2}{l}##

    Now I can express v from this equation and place it into (1). Is that correct?
     
  2. jcsd
  3. Aug 3, 2015 #2

    SammyS

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    Yes. That looks to be correct.
     
  4. Aug 3, 2015 #3
    Please check if my final answer is correct.

    ##mv^2-Bqlv-mgl=0##
    ##v=\frac{Bql+\sqrt{B^2q^2l^2+4m^2gl}}{2m}##


    ##v_o=\sqrt{4lg+\frac{Bql+\sqrt{B^2q^2l^2+4m^2gl}}{2m}}##
     
  5. Aug 3, 2015 #4

    TSny

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    I think this expression for ##v## is correct.

    Did you forget to square ##v## when you substituted for ##v## inside the radical?
     
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