Page 1 of 3 123 LastLast
Results 1 to 15 of 33

Math Help - Modeling a Mass-Spring System

  1. #1
    Junior Member
    Joined
    May 2011
    Posts
    54

    Modeling a Mass-Spring System

    A particle of mass
    m is suspended from a ceiling by a spring of natural length l and

    stiffness
    k. Assume that there is no damping

    If the mass is displaced from equilibrium show that the equation governing its
    subsequent motion is given by
    mz'' + kz = 0.

    My working:If we let e be the extension then N2L is ke-mg=0 so ke=mg.

    An engineer measures the mass of the particle to be
    m = 2, and furthermore k = 4.
    He also finds that internal friction in the spring provides damping that is proportional

    to the velocity of the particle with a constant of proportionality
    μ = 4.

    Show that the equation for the subsequent motion after the displacement of the
    mass is now given by
    2z''
    + 4z' + 4z = 0

    My working: what is the constant of proportionality?


    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    For the first problem, I'm not sure your application of N2L is correct. The first step is to assign a coordinate system. I believe z is intended to be the displacement from equilibrium. Do you want z to be positive up or down?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2011
    Posts
    54
    up please lol. My reasoning was that the restoring force and the displacement act in opposite directions.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    Quote Originally Posted by boromir View Post
    up please lol. My reasoning was that the restoring force and the displacement act in opposite directions.
    They do. So what are all the forces acting on this spring?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    May 2011
    Posts
    54
    ke and mg
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    Quote Originally Posted by boromir View Post
    ke and mg
    What's e? In particular, how does e relate to z?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    May 2011
    Posts
    54
    yeah e=z
    Follow Math Help Forum on Facebook and Google+

  8. #8
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    So what does Newton's Second Law say? Where's the acceleration term in your equations?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    May 2011
    Posts
    54
    acceleration =0 since it's in equilibrium. You may have missed that. It's the next part i'm stuck
    Follow Math Help Forum on Facebook and Google+

  10. #10
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    Quote Originally Posted by boromir View Post
    acceleration =0 since it's in equilibrium. You may have missed that. It's the next part i'm stuck
    Incorrect. It's displaced from equilibrium. It is not actually in equilibrium. Another clue: "its subsequent motion". This mass is moving!
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Junior Member
    Joined
    May 2011
    Posts
    54
    ok so the acceleration is z'' so I get kz-mg=mz''
    Follow Math Help Forum on Facebook and Google+

  12. #12
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    Much closer. However, there are still two problems.

    1. z is measured from the equilibrium point of the mass on the spring, whereas the equation you wrote is really more correct for a different variable y, measured from the equilibrium point of the spring without a mass on it. This point is getting at the heart of the problem here: what's the difference between the system with a mass and without a mass? Another way of phrasing it is this: what is the effect of gravity on such a vertical mass-spring system?

    2. The kz term should have the opposite sign (in addition to the change mentioned above), since the spring is a restoring force.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Junior Member
    Joined
    May 2011
    Posts
    54
    I don't know, I suppose you get an mg term. Come on, spill the goods.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    Quote Originally Posted by boromir View Post
    I don't know, I suppose you get an mg term. Come on, spill the goods.
    Well, this is the main point of the problem, as I see it, so I'm not going to just tell you the answer. That's not the way MHF works. I'll give you a hint, though:

    Compare these two scenarios: the spring without any mass in equilibrium, with the mass on the spring in equilibrium. Just imagine hanging the mass on the spring, and waiting until it stops bobbing. As a thought experiment, what's the result going to be?
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Junior Member
    Joined
    May 2011
    Posts
    54
    more extension?
    Follow Math Help Forum on Facebook and Google+

Page 1 of 3 123 LastLast

Similar Math Help Forum Discussions

  1. mass-spring system
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: February 6th 2010, 09:27 AM
  2. Mass-spring system
    Posted in the Differential Equations Forum
    Replies: 9
    Last Post: December 20th 2009, 11:16 AM
  3. Mass-spring system with damping
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: December 19th 2009, 03:31 PM
  4. Mass-Spring System
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: December 19th 2009, 05:46 AM
  5. damped-mass-spring-system
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: December 11th 2009, 07:12 AM

Search Tags


/mathhelpforum @mathhelpforum