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Rolling hoop hit with impulse. Find angle of deflection and max force.

  1. Jul 21, 2014 #1
    1. The problem statement, all variables and given/known data

    A child's hoop of mass [itex]M[/itex] and radius [itex]b[/itex] rolls in a straight line with velocity [itex]v[/itex]. Its top is given a light tap with a stick at right angles to the direction of motion. The Impulse of the blow is [itex]I[/itex].

    a. Show that this results in a deflection of the line of rolling by angle [itex]\phi = I/Mv [/itex], assuming that the gyroscope approximation holds and neglecting friction with the ground.

    b. Show that the gyroscope approximation is valid provided [itex] F << \dfrac{2Mv^2}{b}[/itex], where F is the peak applied force.

    3. The attempt at a solution

    I don't know for sure the direction of the impulse but assuming it is to the side, that would be the 'angular impulse' would be [itex]Ib[/itex] in the direction perpendicular to the angular momentum [itex]Mbv[/itex] of the wheel, so [itex]\tan\phi = Ib/Mbv = I/Mv ≈ \phi [/itex]. The last part using the small angle approximation.

    For b. I would assume that since [itex]\phi << 1[/itex], therefore [itex] Fdt << Mv [/itex].

    I don't know if any of my attempt is correct.

    Edit: if instead I approximate the impulse by a triangle height [itex]F[/itex] and base [itex] dt [/itex], then the impulse would be [itex]\dfrac{Fdt}{2}[/itex].
     
    Last edited by a moderator: Jul 21, 2014
  2. jcsd
  3. Jul 22, 2014 #2
  4. Jul 22, 2014 #3

    TSny

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    For part a, I think your work is correct.

    For part b, you need to find the meaning of the phrase "gyroscope approximation". I don't think it means ##\phi << 1##. If this is not defined in your text or notes, then try a web search.
     
  5. Jul 23, 2014 #4
    Ok so I checked online as it isn't in my book and it means [itex]\Omega << \omega [/itex] and that they are nearly constant. If I use my expression for impulse along with [itex]\phi << \omega dt = \dfrac{v}{b}dt [/itex] then I get the right answer.

    Thanks for the tip and I hope this is correct.
     
  6. Jul 23, 2014 #5

    TSny

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    OK, good. I think that's right.
     
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