When a mass of 4 kilograms is attached to a spring whose constant is 64 N/m, it comes to rest in the equilibrium position. Starting at t = 0, a force equal to f(t) = 240e−4t cos(4t) is applied to the system. Find the equation of motion in the absence of damping.

    X (t) = 60 cos 4t

Explanation:

When this system is released, a simple harmonic movement began, which is described by the equations

           x = A cos (wt + fi)

Angular velocity is

            w = ra K / m

            w = RA 64/4

            w = 4 rad / s

To find the phase angle (fi) we use the condition that gives leaves of rest for x = 0

       Let's find the speed

              v = dx / de

              v = A w sin (wt + fi)

              0 = A w sin (0+ fi)

    Therefore the value of fi = 0

To find the amplitude let's use the acceleration of the system

             .a = dv / dt

              .a = A w2 cos wt

              If we use Newton's second law

              F = m a

               240 e -4t cos 4t = a w2 cos wt

At the initial time t = 0

                240 = a w2

                .a = 240 / w2

                 .a = 240/42

                  .a = 60

The equation remains

               X (t) = 60 cos 4t