1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cantilever Investigation

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

    1.Hi,

    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

    and:

    h= 4MgL^3/Ebd^3

    where:
    b= width of cantilever
    d= thickness of cantilever
    E= Youngs Modulus
    M=Mass
    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?

    Thanks.

    ash.

    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
      h.pdf
      File size:
      40.1 KB
      Views:
      154
    • T.pdf
      T.pdf
      File size:
      41.1 KB
      Views:
      136
  2. jcsd
  3. Dec 14, 2007 #2

    rock.freak667

    User Avatar
    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

    [tex]s=\frac{Wl^3}{3IE}[/tex]

    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

    [tex]M\frac{d^2s}{dt^2}=\frac{-3IE}{l^3}s[/tex]

    OR
    [tex]\frac{d^2s}{dt^2}+\frac{3IE}{Ml^3}s=0[/tex]

    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:
    [tex]I=\frac{bd^3}{12}[/tex]



    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

    rock.freak667

    User Avatar
    Homework Helper

    Well from
    [tex]M\frac{d^2s}{dt^2}=\frac{-3IE}{l^3}s[/tex]

    [tex]\frac{d^2s}{dt^2}=\frac{-3IE}{Ml^3}s[/tex]

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

    so from that

    [tex]\omega=\sqrt{\frac{3IE}{Ml^3}}[/tex]

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

    [tex]T=\frac{2\pi}{\omega}[/tex]
     
  6. Dec 15, 2007 #5
    Oh yeah I see it now, Thanks alot.

    Ash.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Cantilever Investigation
  1. Investigation Homework (Replies: 16)

  2. Friction Investigation (Replies: 4)

Loading...