Answer to Question #286416 in Classical Mechanics for Shah

Question

An Over Damped harmonic oscillator satisfied the equation (d²x/dt²)+(10dx/dt)+16x=0. At time t=0 the particle is projected from the point x=1 towards the origin with the speed "U". Find x(t) in subsequent motion.

Answer 1

The equation of motion of overdamped oscillator is given by

$$\frac{d²x}{dt²}+10\frac{dx}{dt}+16x=0$$

Let

$$x=e^{\lambda t}$$

The characteristic equation:

$$\lambda^2+10\lambda+16=0$$

Roots:

$$\lambda_1=-8,\quad \lambda_2=-2$$

Solution:

$$x(t)=C_1e^{-8t}+C_2e^{-2t}$$

The initial conditions

$$x(0)=1,\quad x'(0)=-u$$

give

$$x(0)=C_1+C_2=1\\ x'(0)=-8C_1-2C_2=-u$$

Finally, we get

$$x(t)=\frac{1}{3}\left(u/2-1\right)e^{-8t}-\frac{1}{3}\left(u/2-4\right)e^{-2t}$$