Difference between revisions of "Hooke's Law"
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The force <math> \vec F = -k \vec x</math> can be interpreted as a vector field depending on the displacement vector <math>\vec x</math>. Since this field is the gradient of the function <math>\frac{1}{2} k|x|^2</math>, this means that <math>\vec F</math> is a [[conservative field]]. As a result, <math>E = \frac{1}{2} kx^2</math> is conserved, and is the potential energy stored in the string. Moreover, <math>E</math> depends only on <math>\vec x</math>, and not the path that the particle takes to get to <math>\vec x</math>, and so it is said to be a [[path independent]] quantity. | The force <math> \vec F = -k \vec x</math> can be interpreted as a vector field depending on the displacement vector <math>\vec x</math>. Since this field is the gradient of the function <math>\frac{1}{2} k|x|^2</math>, this means that <math>\vec F</math> is a [[conservative field]]. As a result, <math>E = \frac{1}{2} kx^2</math> is conserved, and is the potential energy stored in the string. Moreover, <math>E</math> depends only on <math>\vec x</math>, and not the path that the particle takes to get to <math>\vec x</math>, and so it is said to be a [[path independent]] quantity. | ||
| − | + | It should be noted that in reality, Hooke's law is merely an approximation, and no physical spring actually has precisely this behavior. However, for most materials a version of Hooke's law holds for reasonable ranges of <math>\vec x</math>. This is called the ''elastic range'' of the material. | |
[[Category:Physics]] | [[Category:Physics]] | ||
Revision as of 23:12, August 2, 2010
Hooke's Law states that the force exerted by a stretched or compressed spring is a restoring force and is linearly proportional to the distance that the spring is stretched or compressed beyond its relaxed length. The formula for Hooke's law is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec F = -k \vec x } where k is the spring stiffness constant which has units of newtons per meter and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec x } is the displacement from the equilibrium position.
The negative sign here is to indicate that the force is in the direction opposite the displacement. For example, if is to the right, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -k \vec x} is to the left, indicating that the mass is pulled back towards the spring.
The force can be interpreted as a vector field depending on the displacement vector . Since this field is the gradient of the function , this means that is a conservative field. As a result, is conserved, and is the potential energy stored in the string. Moreover, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} depends only on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec x} , and not the path that the particle takes to get to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec x} , and so it is said to be a path independent quantity.
It should be noted that in reality, Hooke's law is merely an approximation, and no physical spring actually has precisely this behavior. However, for most materials a version of Hooke's law holds for reasonable ranges of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec x} . This is called the elastic range of the material.