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Homework Help: Dimensional analysis and frequency

  1. Aug 29, 2004 #1
    dimensional analysis (Please help as soon as possible!)

    The Space Shutte astronauts use a massing chair to measure their mass. The chair is attached to a spring and is free to oscillate back and forth. The frequency of the oscillation is measured and that is used to calculate the total mass m attached to the spring. If the spring constant k is measured in kg/s^2 and the chairs frequency f is .50s^-1 for a 62-kg astronaut, what is the chair's frequency for a 75-kg astronaut? The chair itself has a mass of 10.0 kg. [Hint: use dimensional analysis to find out how f depends on m and k.]

    Could you please explain how to work this problem step by step, because i have no clue how to even begin, thank you.
     
    Last edited: Aug 29, 2004
  2. jcsd
  3. Aug 29, 2004 #2

    Fredrik

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    We know that k is measured in kg/s^2, m is measured in kg, and f is measured in 1/s. I would write this as

    [tex][k]=MT^{-2}[/tex]
    [tex][m]=M[/tex]
    [tex][f]=T^{-1}[/tex]

    (M=mass, T=time)

    We want to find how f depends on k and m. We do this by solving the equation

    [tex][f]=[k]^a[m]^b[/tex]

    for a and b. Using what we know about the units, this equation takes the form

    [tex]T^{-1}=M^aT^{-2a}M^b[/tex]

    The solution is obviously

    [tex]a=\frac{1}{2}[/tex]
    [tex]b=-\frac{1}{2}[/tex]

    so we know that

    [tex]f=A\sqrt\frac{k}{m}=\sqrt\frac{A^2k}{m}[/tex]

    where A is a dimensionless constant.

    We can solve this equation for A^2k:

    [tex]A^2k=mf^2[/tex]

    Now you can calculate A^2k using the numbers f=0.5 and m=72. Then insert the result along with m=85 into the formula for f above.

    As you see, you don't need to know A and k separately. It's enough to know A^2k. If you would like to know what A is I can tell you that it's 1/(2pi), but you can't get that from dimensional analysis. You would have to solve the equation of motion (Newton's second law) to get that.
     
  4. Aug 29, 2004 #3
    Thanks for the help.
     
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