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An advanced two spring weighing machine

  1. Jan 9, 2015 #1
    1. The problem statement, all variables and given/known data
    A more sophisticated weighing machine contains two springs. The effect of these is that the machine is more sensitive to masses less than some fixed value M0 than it is to greater masses. The equilibrium displacement is alpha1*M when M<M0, and (alpha2+c) when M≥M0 , where alpha1, alpha2 and c are constants.

    An object of mass M0 is held in contact with the pan at x=0 and released at time t=0 .

    Find an expression for the period of the subsequent motion, and find the period of the oscillations if the object mass M0=450g, and the values of the constants are: alpha1=0.800mkg−1 and alpha2=0.650mkg−1

    Find an expression for the maximum reading of the pointer in terms of M0, alpha1 and alpha2. Calculate the maximum reading using M0=450g, alpha1=0.800mkg−1 and alpha2=0.650mkg−1

    2. Relevant equations
    F=-kx
    SHM equation: mx''+kx=0

    3. The attempt at a solution
    The displacement has to be continuous when M=M0, so we can derive an expression for c:
    alpha1*M0 = alpha2*M0+c
    c= (alpha1-alpha2)*M0

    I have been trying to figure out what the two spring system is - I tried two vertical springs in series with the pan and mass between them and I tried two springs in parallel, but neither reproduces the displacement behaviour of the problem. It's not clear to me if I should be trying to understand the setup of the system or they want me to solve a general case. Either way, I'm stumped.
     
  2. jcsd
  3. Jan 9, 2015 #2

    ehild

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    Welcome to PF!
    I think a picture of the set-up would be necessary. Is not there one somewhere around the problem text?
     
  4. Jan 10, 2015 #3

    haruspex

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    You don't really need to understand the structure. You all the information about how it behaves. A possible design is that with no load only one spring is in contact with the pan. When the pan is depressed by alpha1*M0 it makes contact with the second spring.
     
  5. Jan 10, 2015 #4

    ehild

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    Very wise! Can it be this?
    twospringbalance.JPG
     
  6. Jan 10, 2015 #5

    haruspex

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    Yes, that's what I had in mind, thanks.
     
  7. Jan 11, 2015 #6
    Thank you so much, that makes sense now! I think I was trying to make it more complicated than it needed to be.
     
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