a) To graph the equation of motion for the mass spring system with damping, we can use a graphing calculator. The graph will show the relationship between time (t) and displacement (y) of the mass. This graph will allow us to visualize how the mass oscillates over time.

The equation of motion is given by y” + 8y’ + 2y = 0, where y denotes the displacement of the mass and t represents time. This is a second-order linear homogeneous ordinary differential equation. To solve this equation, we can assume a solution of the form y = e^(rt), where r is a constant.

Substituting this into the equation, we get r^2 e^(rt) + 8r e^(rt) + 2e^(rt) = 0. Dividing by e^(rt) and factoring out e^(rt), we obtain the characteristic equation r^2 + 8r + 2 = 0. This quadratic equation can be solved using the quadratic formula. The solutions for r are complex numbers, which indicate that the equation of motion represents an underdamped system.

The general solution for the displacement y(t) is y(t) = C1e^(r1t) + C2e^(r2t), where C1 and C2 are constants and r1 and r2 are the solutions of the characteristic equation. In this case, since the solutions are complex, we have r1 = -4 + √14i and r2 = -4 – √14i. Therefore, the general solution becomes y(t) = C1e^((-4 + √14i)t) + C2e^((-4 – √14i)t).

To graph the equation of motion on a calculator, we can choose specific values for the constants C1 and C2 and plot the resulting function y(t). The graph will show the oscillatory behavior of the mass with respect to time.

b) The best description of the system represented by the mass spring equation of motion y” + 8y’ + 2y = 0 is underdamped. An underdamped system refers to a system that exhibits oscillatory behavior with decreasing amplitude over time.

In our case, the system is underdamped because the solutions for r in the characteristic equation are complex numbers. Complex solutions indicate that the motion of the mass will oscillate with a certain frequency and decay over time. This decay in amplitude is due to the damping term in the equation, which is represented by the coefficient 8 in front of the first derivative term y’.

The underdamped nature of the system can also be observed by examining the general solution. As mentioned earlier, the general solution includes terms of the form e^(rt), where r is a complex number. These exponential terms represent oscillatory behavior in the displacement y(t). The presence of complex solutions and oscillatory behavior indicates that the mass spring system is underdamped.

It is important to note that overdamped and critically damped systems would have different characteristic equations with real solutions. An overdamped system refers to a case where the system has slow decay with no oscillations, while a critically damped system exhibits rapid convergence to equilibrium without oscillations. In the given equation of motion, the complex solutions for r rule out these possibilities.

Therefore, the system represented by the equation of motion y” + 8y’ + 2y = 0 is best described as underdamped. The graphing calculator can visually illustrate this behavior by displaying the oscillatory motion of the mass with decreasing amplitude over time.