A 1-kilogram mass is attached to a spring whose constant is 14 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 9 times the instantaneous velocity. Determine the equations of motion if the following is true. (a) the mass is initially released from rest from a point 1 meter below the equilibrium position. (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 13 m/s.

Part(a): The equation of motion is .

Part(b): The equation of motion is  .

Explanation:

If '' be the mass of the object, '' be the force constant and '' be the damping constant, then the equation of motion of the particle can be written as

Given = 1 Kg, = 14 , = 9. Substituting these values in equation (I),

Taking a trial solution , the auxiliary equation can be written as

and its solutions are , resulting the general solution

The velocity at any instant of time of the mass is

Part(a):

Given . Substituting these values in equation (III) and (IV),

Solving equations (V) and (VI), we have

So the equation of motion is

Part(b):

Given . Substituting these values in equation (III) and (IV),

Solving equations (V) and (VI), we have

So the equation of motion is

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