The mass is acted on by an external force of 10 sin(t/2)N (newtons) and moves in a medium that imparts a viscous force of 2 N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value problem describing the motion of the mass.

u''(t) + 10*u'(t) + 98*u(t) = 2 sin(t/2) , u(0) = 0 , u'(0) = 0.03 m/s

Explanation:

Solution:

- The Force exerted F(t) = 10 sin(t/2)

- The viscous force at time t* is given by F(t*) = - 2N

- The speed of mass m at point t* is u'(t*) = 0.04 m/s

- The initial velocity u(0) = 0.03 m/s

- The IVP ODE of a damped system is given by a general form as follows:

                m*u''(t) + b*u'(t) + k*u(t) = F

- We will evaluate the system stiffness constant k and damping constant b for the system.

- We know that system stiffness k is given by Newton's equation of motion at equilibrium position we have:

                                Fs = Fg

Where, Fs: spring force = k*x

            Fg: Weight of the object = m*g

                                k*x = m*g

                                k = m*g / x

- Given mass m = 5 kg stretches spring by x = 10 cm (Missing part of question)

- The stiffness constant k is:

                                k = 5*9.81 / 0.1

                                k = 490 N/m

- The damping force at time t* is given by:

                                Fd(t*) = - b*u'(t*)

                                b = - Fd(t*) / u'(t*)

                                b = 2 / 0.04

                                b = 50 Ns/m

- The IVP for the system is given as:

                   m*u''(t) + b*u'(t) + k*u(t) = F

                   5u''(t) + 50*u'(t) + 490*u(t) = 10 sin(t/2)

                   u''(t) + 10*u'(t) + 98*u(t) = 2 sin(t/2) , u(0) = 0 , u'(0) = 0.03 m/s