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zenterix
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- Homework Statement
- I'd like to understand a simple derivation I found in a book that aims to show that a conservative 1d force is approximately linearly related to position.
- Relevant Equations
- The derivation uses a Taylor series of a potential function for the force.
There are a few statements and calculations which I don't fully understand.
Everything below is from this (free) book about Vibrations and Waves from MIT OCW's 8.03 course.
Consider a conservative force ##F=-V'(x)##.
At a point of equilibrium, ##x_0##, the force vanishes.
$$F=-V'(x_0)=0$$
Let ##x_0=0##. We can write this because we are free to redefine the coordinate system.
Expand the force in Taylor series
$$F(x)=-V'(x)=-V'(0)-xV''(0)-\frac{1}{2}x^2V'''(0)+\ldots\tag{1}$$
$$=xV''(0)-\frac{1}{2}x^2V'''(0)+\ldots\tag{2}$$
The equilibrium is stable if ##V''(x_0)>0## as this means ##x_0## is a local minimum of potential energy.
For sufficiently small $x$, the second term in (2), and all subsequent terms will be much smaller than the first term in (2).
The third term is negligible if
$$|xV'''(0)|<<V''(0)\tag{3}$$
Then, (3) becomes
$$x<<L\tag{4}$$
My questions are
1) How do we derive (3) exactly?
2) What does the quote above about the factor ##1/L## mean?
3) I think with answers to 1) and 2) I can answer this: where does (4) come from?
The point of the derivation above is to show that for small displacements from equilibrium, we obtain an approximately linear relationship between force and displacement, ie we get Hooke's law.
Consider a conservative force ##F=-V'(x)##.
At a point of equilibrium, ##x_0##, the force vanishes.
$$F=-V'(x_0)=0$$
Let ##x_0=0##. We can write this because we are free to redefine the coordinate system.
Expand the force in Taylor series
$$F(x)=-V'(x)=-V'(0)-xV''(0)-\frac{1}{2}x^2V'''(0)+\ldots\tag{1}$$
$$=xV''(0)-\frac{1}{2}x^2V'''(0)+\ldots\tag{2}$$
The equilibrium is stable if ##V''(x_0)>0## as this means ##x_0## is a local minimum of potential energy.
For sufficiently small $x$, the second term in (2), and all subsequent terms will be much smaller than the first term in (2).
The third term is negligible if
$$|xV'''(0)|<<V''(0)\tag{3}$$
Typically, each extra derivative will bring with it a factor of ##1/L## where ##L## is the distance over which the potential energy changes by a large fraction.
Then, (3) becomes
$$x<<L\tag{4}$$
My questions are
1) How do we derive (3) exactly?
2) What does the quote above about the factor ##1/L## mean?
3) I think with answers to 1) and 2) I can answer this: where does (4) come from?
The point of the derivation above is to show that for small displacements from equilibrium, we obtain an approximately linear relationship between force and displacement, ie we get Hooke's law.
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