A mass weighing 3232 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At time tequals=0, an external force of Upper F (t )F(t)equals=3 cosine 5 t3cos5t lb is applied to the system. If the spring constant is 1515 lb/ft and the damping constant is 33 lb-sec/ft, find the steady-state solution for the system. Use g equals 32 ft divided by sec squaredg=32 ft/sec2.

The steady state  solution is    

Step-by-step explanation:

From the the question we are told that

     The mass is

       The force applied is

       The spring constant is

        The damping constant is

        The acceleration due to gravity is

Generally the model differential equation for the motion of the mass attached to a string is mathematically represented as

Substituting values

Substituting given value for F(t)

Applying the method of undermined coefficient in order to obtain the steady state solution

   Let

            be the steady state solution

it means that

And

Now substituting this into the equation

Comparing coefficients  on both sides

                   Right Hand Side         Left Hand Side          

   Cos (5t)       3                                    11A + 6B

    Sin (5t)       0                                       11B -6A

 From above we have that

And    

From equation one we see that

Substituting this for B in equation 1

Multiply through by 11

substituting for A in equation 3

So the steady state solution is

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