Spring Energy

The force exerted by a spring is a conservative force. If the spring is linear, then its resisting force is given by $f=-kx$. The potential energy of a linear spring is then:

$$P(x)=\frac{1}{2}k{x}^2$$

where $x$ is the deflection from the free length of the spring. The kinetic energy due to movement is $\frac{1}{2}m{v}^2$, where $v=\dot{x}$.

A torsional spring exerts a moment $M$ if it is twisted, which is given by $M=k_t\theta$, where $\theta$ is the twist angle. The work done by this moment and stored as potential energy in the spring is:

$$P(\theta )={\int }_0^{\theta }{k}_t\theta d\theta =\frac{1}{2}{k}_t{\theta }^2$$

The conservation of energy principles states that $\Delta \text{PE}+\Delta \text{KE}=0$, or $\text{PE}+\text{KE}=\text{constant}$.