Question

Derive expressions for potential energy and kinetic energy of an oscillating spring-mass
system.

Answer 1

Full energy of spring-mass system is $W = Wp + Wk$ ;

$Wp = \frac{kx^2}{2} -$ Potential energy

$Wk = \frac{mV^2}{2}$ - kinetic energy

x - displacement of mass;

K - constant factor characteristic of the spring, its stiffness.

m-mass;

V-velocity of mass;

If equation of oscillation is $x = A\cos (wt \pm \varphi)$ or $x = A\sin (wt \pm \varphi)$ , we have

$Wp = \frac{k\left(\left(A\cos(wt\pm\varphi)\right)^{2} \right.}{2} = \frac{k}{2} A^{2}\cos^{2}(wt\pm\varphi)$

$V = x^{\prime}(t)$ - derivative from $\mathbf{x}(t)$

$x^{\prime}(t) = \left(Asin(wt\pm \varphi)\right)^{\prime} = A\cdot w\cdot \cos (wt\pm \varphi)$

$Wk = \frac{mV^2}{2} = \frac{m}{2}\cdot A^2\cdot w^2\cdot sin^2 (wt\pm \varphi)$

$W = \frac{m}{2} \cdot A^2 \cdot w^2 \cdot \sin^2(wt \pm \varphi) + \frac{k}{2} A^2 \cos^2(wt \pm \varphi)$

Where $w = \sqrt{k / m}$