Physics: Problems and Solutions

Hooke's law

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Hooke's law is a law which states that the force, F, required to bend a spring (or some other elastic object) is directly proportional to the distance X by some constant k, known as the stiffness constant.


This is sometimes written as


In this case, F is equal to the force with which the spring pushes back.

By integrating with respect to x, we can find the work needed to compress or stretch a spring a given distance and the potential energy stored in said spring.

W = E_p = \frac{1}{2} k x^2

Hooke's law is only an approximation, as all materials will deform past a certain point (called the elastic limit). In fact, many objects deviate from Hooke's law well before their elastic limits. However, for most cases, Hooke's law is fairly accurate.

Harmonic motionEdit

If a weight is attached to a spring and the spring is stretched or compressed released, the motion can be described as

x=A \cos ( \sqrt{ \tfrac{k}{m}} t )

where A is the amplitude, or how far the spring is stretched, k is the stiffness constant, and m is the mass of the weight. By taking the first and second derivatives, the speed and acceleration can be found.

The total energy of the system is equal to

E = \frac{1}{2} k A^2

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