(a)In the mechanical system shown in Figure P2.62(a), the force x(t) applied to the mass...

(a)In the mechanical system shown in Figure P2.62(a), the force x(t) applied to the mass represents the input, while the displacement y(t) of the mass represents the output. Determine the differential equation relating x(t) and y(t). Show that the natural response of this system is periodic

(b) Consider Figure P2.62(b), in which the force x(t) is the input and the velocity y(t) is the output. The mass of the car is m, while the coefficient of kinetic friction is p. Show that the natural response of this system decays with increasing time.

(c) In the mechanical system shown in Figure P2.62(c), the force x(t) applied to the mass represents the input, while the displacement y(t) of the mass represents the output.

(i) Determine the differential equation relating x(t) and y(t).

(ii) Show that the homogeneous solution of the differential equation from part (i) has the form and specify the value of a.

(iii) Show that, since the force and displacement are restricted to be real, the natural response of the system is a decaying sinusoid.