We developed a program that calculates the actual velocity of a mass on a linear spring.
[math]
\def\MYdef{\mathrel{\stackrel{\rm def}=}}
\def\({\left(}
\def\){\right)}
[/math]
The Hooke's law has the form:
- [math]
F = - Cu,
[/math]
- [math]
m \ddot u + Cu = 0.
[/math]
Dividing by [math]m[/math] and multiplying by [math]\dot u[/math] we arrive at:
- [math]
\ddot u \dot u + \omega^2 \dot u u = 0, \quad \omega_0 \MYdef \sqrt{\frac{C}{m}},
[/math]
- [math]
\dot u^2 + \omega^2 u^2 = 0, \quad \(x^2(t)\)'_t = 2x(t)\dot u(t),
[/math]
- [math]
\(\frac{\dot u}{\omega}\)^2 + u^2 = 0.
[/math]
By introducing notation
- [math]\frac{\dot u}{\omega} = y, \quad u = x[/math].
We obtain:
[math] y^2 + x^2 = 0 [/math].
Developers: D.V. Tsvetkov, A.M. Krivtsov.
