Main Concept

Hooke's Law states that the force $F$ exerted by a stretched spring is proportional to its displacement, $x$, from the equilibrium position. The constant of proportionality, $k$, is known as the spring constant:

$F\=-k x$.

If an object of mass $m$ is attached to the end of the spring, extended and then released (possibly with a nonzero initial velocity), it will oscillate periodically according to the formula:

$x\left(t\right) \= A\cdot \sin \left(\sqrt{\frac{k}{m}}t\+\mathrm{\ϕ}\right)\,$

where the amplitude $A$ and phase $\mathrm{\ϕ}$ depends on the initial velocity and position of the mass at the time of release. This system is called the simple harmonic oscillator, and the associated motion is called simple harmonic motion.

Derivation

By combining Newton's second law $F\=m a$  with Hooke's law and noting that the acceleration $a$ is just $\frac{{\ⅆ}^{2}x}{\ⅆt^2}$, we obtain   $m\frac{{\ⅆ}^{2}x}{\ⅆt^2} \=-k x$. The solution of this differential equation for $x\left(t\right)$ can be expressed as $x\left(t\right) \= A\cdot \sin \left(\sqrt{\frac{k}{m}}\cdot t\+\mathrm{\ϕ}\right)$,   where $A$is is the amplitude of the oscillation, i.e. its maximum displacement, and $\mathrm{\ϕ}$ is the initial phase. In the animations below, we have set  $A\=1$and $\mathrm{\ϕ}\=0\,$thus   $x\left(t\right) \= \sin \left(\sqrt{\frac{k}{m}}\cdot t\right)\.$ $$

Spring Constant, $\mathit{k}$ 

 Mass of particle, $\mathit{m}$ 

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