Mass weighing 16 pounds stretches a spring 8 3 feet. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 1 2 the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t)

The answer is

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Explanation:

Taking into consideration a volume weight = 16 pounds originally extends a springs feet but is extracted to resting at 2 feet beneath balance position.

The mass value is =

The source of the hooks law is stable,

Number  times the immediate speed, i.e .. Damping force

The m^2+m+12=0 and m is an auxiliary equation,

Therefore, additional feature

Use the form of uncertain coefficients to find a particular solution.  

Assume that solution equation,

These values are replaced by equation ( 1):

Going to compare cos3 t and sin 3 t coefficients from both sides,  

The cost3 t is 3A + 3B= 20 coefficients  

The sin 3 t is 3B -3A = 0 coefficient  

The two equations solved:

Replace the very first equation with the meaning,

equation is

The ultimate plan for both the equation is therefore

Initially, the volume of rest x(0)=2 and x'(0) is extracted by rest i.e.  

Throughout the general solution, replace initial state x(0) = 2,

Replace x'(0)=0 with a general solution in the initial condition,

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