Final answer:
A 10 kg weight is attached to a spring with constant k = 160 kg/m and subjected to an external force F(t) = 360 sin(2t). The weight is initially displaced 3 meters above equilibrium and given an upward velocity of 3 m/s ts displacement for t > 0, y(t) = 3cos(4t) +9/4sin(4t) + 3/4t.
Step-by-step explanation:
The displacement equation y(t) was derived by first finding the complementary function, considering the initial conditions, and then incorporating the particular integral derived from the external force (F(t). The complementary function, (
(t), encompasses the homogenous solution of the differential equation by considering the characteristics of the undriven, undamped harmonic oscillator. It leads to
(t) = Acos(4t) + B sin(4t) after considering the initial displacement and velocity.
Next, the particular integral,
(t), was determined by assuming a solution form according to the external force F(t). Using the method of undetermined coefficients,
(t) was found to be 3/4t. By adding the complementary function and the particular integral, the general solution was formulated as y(t) =
(t) +
(t).
The coefficients were determined by matching terms and ensuring the solution satisfied both the differential equation and the initial conditions of displacement and velocity. This resulted in the final equation y(t) = 3cos(4t) + 9/4sin(4t) + 3/4t, which represents the displacement of the 10 kg weight attached to the spring for t > 0, with positive displacement measured upwards from the equilibrium position.