Damped Vibrations

I will briefly lay out the steps of the problem.

• Solve the following differential equation for $x$ as a function of time: $M\ddot{x} + c\dot{x} + kx = 0$ , where, $x(0) = 10$ and $\dot{x}(0) = 0$ .
• Having obtained the particular solution $x(t)$ , differentiate with respect to time to obtain $\dot{x}(t)$ .
• The total energy of the system is the sum of the kinetic energy of the mass and potential energy stored in the spring: $E = \frac{1}{2}kx^2 + \frac{1}{2}M\dot{x}^2$

Differentiating the energy with respect to time: $\frac{dE}{dt} = \dot{x}(kx + M\ddot{x})$ .

We know that: $M\ddot{x} + c\dot{x} + kx = 0$ . This implies: $M\ddot{x} + kx = -c\dot{x}$

Replacing the above result in the 'rate of energy change' equation gives us: $\frac{dE}{dt} = -c\dot{x}^2$

• Replacing $\dot{x}$ with its particular solution, gives us rate of energy change as a function of time. The negative sign indicates that energy is being dissipated from the system. Substitute $t = 2$ and obtain the answer.

I am, for now, refraining to answer the bonus questions posed. These are conceptual questions worth thinking about.