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Why x^2 for PE?

  1. Jul 29, 2013 #1
    Hi all,

    I notice the patterns such as v = 0.5at^2 and PE(spring)=0.5kx^2, etc...but, all examples I have seen show the slope (acceleration, or the spring constant, k) as being a 45 degree angle. Thus, the area of the triangle underneath the graph makes good sense (x^2 or t^2).

    But, let's say we have an example where the spring constant is much larger or much smaller. Why is x^2 still valid as the base x height, if the two are not equal values?
  2. jcsd
  3. Jul 29, 2013 #2
    Maybe I'm just tired, but I'm really confused about your question. So that people don't start answering every question except what you meant, would you mind clarifying? (also, x, not v in your first one)
    What do you mean a or k is a 45 degree angle?
    I'm guessing that the answer you are looking for is going involve integration. What level of math are you comfortable with?
  4. Jul 29, 2013 #3
    Sorry for the confusion. I'm evidently tired too.

    I have it answered. All it took was writing PE = 0.5 k(x) times x. I couldn't intuitively see the x^2 mentally.
  5. Jul 29, 2013 #4


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    Staff: Mentor

    Those terms come from calculus derivations. Are you familiar yet with differential and integral calculus?
  6. Jul 29, 2013 #5


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    Science Advisor
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    Gold Member

    that is s = 1/2at^2
    when you plot v vs.t, and the acceleration is constant, then you have a linear equation v = at, and the slope of the line is the acceleration. Thus, the area of the triangle underneath the graph is 1/2 at^2. Or since F=kx, the slope of the line is k. and the PE is 1/2kx^2. But the slope is not always 45 degress, it could be much higher , say 60 degrees , but the area under the curve (straight line) is still the same, the area of the triangle.
    Tey don't have to be equal. The area of the triangle is still 1/2kx^2 for the spring PE case, whether k is 1 (straight line graph for f = kx, k=1, 45 degree slope, or k is greater than 1 (higher slope and thus greater angle) or less than 1, (less steep slope , angle less than 45 degrees).
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