Suppose a spring with spring constant $\displaystyle{\ 6 \ \text{N/m} \ }$ is horizontal and has one end attached to a wall and the other end attached to a $\displaystyle{\ 2 \ \text{kg} \ }$ mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is $\displaystyle{\ 1 \ \text{N $\cdot$ s/m} \ }$.

• Set up a differential equation that describes this system. Let $\ x \ $ denote the displacement, in meters, of the mass from its equilibrium position, and give your answer in terms of $\displaystyle{\ x, x^{\,\prime}, x^{\,\prime\prime} }$.
  Assume that positive displacement means the mass is farther from the wall than when the system is at equilibrium.
  equation

• Find the general solution for your differential equation.
  Use c1 and c2 to represent arbitrary constants. Use t for the independent variable, to represent the time elapsed in seconds. Your answer should be in the form of an equation: $\displaystyle{\ x = (\ldots)\ }$.
  equation
• Is this system under damped, over damped, or critically damped?
  ? over damped critically damped under damped
• Enter a value for the damping constant that would make the system critically damped.
  $\displaystyle{\ \text{N $\cdot$ s / m}}$