Hooke's Law

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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  \vec F = -k \vec x where k is the spring stiffness constant which has units of newtons per meter and  \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 \vec x is to the right, then -k \vec x is to the left, indicating that the mass is pulled back towards the spring.

The force  \vec F = -k \vec x can be interpreted as a vector field depending on the displacement vector \vec x. Since this field is the gradient of the function \frac{1}{2} k|\vec x|^2, this means that \vec F is a conservative field. As a result, E = \frac{1}{2} k|\vec x|^2 is conserved, and is the potential energy stored in the string. Moreover, E depends only on \vec x, and not the path that the particle takes to get to \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 \vec x. This is called the elastic range of the material.

If a particle moves only under the influence of the force exerted by a spring, then Newton's second law (\vec F = m \vec a) implies that its displacement satisfies the second-order differential equation m\ddot{\vec x} = - k \vec x. This is solved by \vec x(t) = \vec x_0 \cos(\omega t), where \omega = \sqrt{k/m} is the frequency of oscillation: thus the particle moves in a "sine wave" shape. This is an example of simple harmonic motion.

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