Final answer:
The steady-state solution for a damped harmonic oscillator subjected to an external periodic force can be represented in a form similar to the driving force. However, specifics like amplitude and phase shift require more information, including the spring constant. The general form of the steady-state solution will be periodic and share the frequency with the driving force.
Step-by-step explanation:
We are given a 1-kg mass attached to a spring with an external force F(t) = 2 cos 3t N applied, and the damping constant is 4 N-sec/m. To determine the steady-state solution for the damped harmonic oscillator under a periodic force, we apply the concept of resonance and forced oscillations in physics. The steady-state solution is the particular solution to the differential equation describing the motion which doesn't contain the transient terms that go to zero as time goes to infinity.
For a driven damped harmonic oscillator, the form of the steady-state solution is y(t) = A cos(3t + φ), where A is the amplitude and φ is the phase shift. Both can be found using the method of undetermined coefficients or using a phasor diagram with the given damping coefficient and driving force. However, without the specifics of the spring's natural frequency, which is determined by the spring constant, an exact amplitude and phase cannot be provided.
In the given scenario, important components that are missing to provide an exact answer for the steady-state solution include the spring constant k and the precise form of the damping force (usually assumed to be proportional to the velocity). These would allow us to find the system's natural frequency, compare it to the driving frequency, and then solve for A and φ. Therefore, based on the given information, a general form for the steady-state response can be suggested but not the specific solution. The general form is reflective of the periodic nature of the forced response, which matches the driving frequency of the applied force.