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Homework Help: Cantilever Investigation

  1. Dec 14, 2007 #1
    [SOLVED] Cantilever Investigation


    I am writing a report on cantilever oscillations, my experiment involves fixing different cantilvers e.g. a ruler to the end of a table then measuring the period and height of oscillations while varing the mass attached to the cantilever, and other varients e.g. lenght of cantilever.

    2. I have found these two formulae: (Shown much more clearly in attachments)

    T= 2(pi)*[(4ML^3)/(bd^3E)]^1/2


    h= 4MgL^3/Ebd^3

    b= width of cantilever
    d= thickness of cantilever
    E= Youngs Modulus
    L=Lenght of cantilever
    T=period of oscillations
    h=height of oscillation

    3. I have looked at eqn's involving Hooke's and simple harmonic motion but cannot work out how these formulae have been derived.

    Does anyone know how these formulae where derived, or where I can find information on this in general?



    p.s. I have written out the formulae using math open office and attached them in pdf if it helps make them easier to read.

    Attached Files:

    • h.pdf
      File size:
      40.1 KB
    • T.pdf
      File size:
      41.1 KB
  2. jcsd
  3. Dec 14, 2007 #2


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    Homework Helper

    Well I have done a lab to find the Young's Modulus of a loaded cantilever and this is theory which is written down on the paper:

    The depression,s,due to a load W(=Mg) at the end of a cantilever of length,l, is


    This strain brings into play internal stresses which produce a restoring force equal to W, i.e. equal to [itex]\frac{3IEs}{l^3}[/itex].

    If the acceleration of the load [itex]\frac{d^2s}{dt^2}[/itex] when the cantilever is displaced to produce vertical oscillations,then



    Hence the motion is Simple harmonic and the periodic time,T, is

    [tex]T=2\pi \sqrt{\frac{Ml^3}{3IE}}[/tex]

    from which

    [tex] E=\frac{4\pi^2Ml^3}{3IT^2}[/tex]

    For a beam of rectangular section:

    I hope that helps in some way
    Last edited: Dec 15, 2007
  4. Dec 15, 2007 #3
    Thanks for your reply rock.freak it's really really useful, just one question:

    why does [tex]T=2\pi \sqrt{\frac{Ml^3}{3IE}}[/tex]

    Thanks, ash.
  5. Dec 15, 2007 #4


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    Homework Helper

    Well from


    and that is of the form [itex]a=-\omega^2 s[/itex] where [itex]a=\frac{d^2s}{dt^2}[/itex]

    so from that


    and since it moves with SHM, the period,T, is given by

  6. Dec 15, 2007 #5
    Oh yeah I see it now, Thanks alot.

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