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Tension Distribution/ hooks law

  1. Dec 21, 2011 #1
    A Popsicle stick is easier to break of you apply pressure on its flat surface,however if you change the orientation of the popsicle stick so its no longer flat(popsicle stick is no longer lying on its flat surface but is now lying on its "thinner" surface), it becomes much harder to break.
    (i hope this made sense)

    I've read articles that imply that there is a distribution of tension throughout the Popsicle stick depending on its orientation.

    My Questions are:
    1.Why is the popsicle stick easier to break on its flat surface?
    2.What law of physics explains this?
    3.Can all(most) physical objects be interpreted as a chunk of atoms with spring forces acting on each other?(i've read in physics simulation books that most physics can be modeled by springs and i'm wondering if that's the case for real objects aswell )
     
    Last edited: Dec 21, 2011
  2. jcsd
  3. Dec 22, 2011 #2
    In the simplest model, you can think of the wood as a uniform isotropic material with a definite Young's modulus E and breaking stress σU. "Beam theory" provides a model for the stress distribution in a thin beam made out of such a material. The usual simplifying assumption is that the stress distribution is dominated by the local bending moment and that the shear force distribution can be neglected.

    In that case, the stress is longitudinal (directed along the length of the beam) and varies from compressive on the concave side to tensile on the convex side. If you imagine you are bending the popsicle stick so that the middle bows upward, i.e. is concave down, then the fibers on the top will be in tension and the fibers on the bottom will be in compression. In this ideal model the beam will fail when the maximum tensile strength σU is reached in the fibers on the tensile side.

    The stress at a distance z from the "neutral axis" is

    [itex]\sigma=Mz/I[/itex]

    where M is the bending moment and I is the second moment of area. For a beam of width w and thickness t the second moment is

    [itex]I=wt^3/12[/itex]

    The maximum stress is seen where z is maximum. For a rectangular beam this will be zmax=t/2. So the maximum stress is

    [itex]\sigma_{max}=6M/wt^2[/itex]

    Now suppose the popsicle stick has a rectangular section of width a and thickness b with a >> b. In the ordinary orientation (w=a, t=b) the maximum stress is

    [itex]\sigma_{O}=6M/ab^2[/itex]

    while in the rotated orientation (w=b, t=a) the stress is

    [itex]\sigma_{R}=6M/ba^2[/itex]

    Hence

    [itex]\frac{\sigma_{R}}{\sigma_{O}}=b/a<<1[/itex]

    and the rotated orientation experiences much less stress for the same bending moment. Conversely, it takes a much higher bending moment to reach the ultimate stress in the rotated orientation.

    In actuality, real wood is anisotropic (the grain establishes a preferred direction) and the breaking stress varies from point to point. Common sense will tell you that the wood splinters when it breaks and that the breaking planes are not at all what an idealized model would predict. But the above should be the basic answer to your questions 1 and 2.

    As to question 3, your mileage may vary. Most solids have some linear elastic regime when the strains are small, and the elasticity is due to interatomic forces. I would expect there are substances that have a fairly narrow elastic regime and become non-linear even at low strain levels, but you're welcome to research that yourself. Just remember that a model is just a model.

    BBB
     
  4. Dec 22, 2011 #3
    BBB definitely gave a textbook perfect answer, but considering the language used to phrase the question, I think a less technical answer may be in order.

    I have always thought that the easiest way to consider the fact that the force required to break something changes with the thickness in the direction of breaking is to consider the physicality of mass and to imagine a mass as a solid composed of planes of atoms perpendicular to the breaking force applied.

    Imagine your flat popsicle stick as a bunch of whispy thin layers of atoms (like sheets of paper) stacked together. When you take a stack of papers and fold them, the inner papers stick out and appear to be "longer" while the outer papers appear to be shorter. This is a comparative observation with respect to the middle paper. The outside edge length of the total shape got longer, but the paper on the outside edge stayed the same length, and therefore appeared short. Conversely, the inside edge length of the total shape got shorter, but the paper stayed the same length and so appeared to be too long. The length of the middle of the stack of paper changed lease of all. We can analyze the behavior of a solid during bending in much the same way, saying that when a solid is bent, the side of the solid "outside" of the bending angle gets longer and the "inside" of the solid gets shorter.

    Going back to your popsicle stick, the "layers" of atoms in the popsicle stick cannot move with respect to one another the way sheets of paper can. They behave more like sheets of paper that have all been glued together. In this instance, there is a tendency (and I hope an instinctive understanding) for the papers to want to slip against one another when they are bent, but the glue holds them back and in turn provides a resistant force to the bending. The more papers glued together, the more force is required to bend them. Ergo, bending force (and thus breaking force) increase with thickness.

    A similar way to imagine this, to me, is to imagine the atoms in the solid like marbles (which is usually a horrible model for atoms, but works in this specific instance). Remember that we said a bending solid has a positive change in the outside surface length (the outside surface gets longer) and a negative change in the inside surface length (the inside surface gets shorter). Overall, the volumetric capacity of the shape decreases. This is a geometric fact of the change in shape. However, marbles do not compress and squeeze together. If you have a certain number of marbles in a box, and then you bend the box, you will reach a point where the physical mass of the marbles prevents further bending (further decrease in volume) unless the box is broken, or very simply, two objects having the same mass may not occupy the same physical, 3 dimensional space.

    My roommate and I occasionally tried to disprove this, freshman year. Try as we might, we both could not occupy the same space...
     
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