A$-3-4.$ Obtain the transfer function $x_o\frac{s}{x_i}(s)$ of the mechanical system shown in Figure $3-23($a$).$ Also

obtain the transfer function $E_o\frac{s}{E_i}(s)$ of the electrical system shown in Figure $3-23($b$).$ Show that

these transfer functions of the two systems are of identical form and thus they are analogous systems.

EXAMPLE $5\mathrm{.}2$ Design and analysis of a sprue for casting

The desired volume flow rate of the molten metal into a mold is $0\mathrm{.}01\frac{m^3}{m}in.$ The

top of the sprue has a diameter of $20$ mm and its length is $200$ mm $.$ What diame$-$

ter should be specified at the bottom of the sprue in order to prevent aspiration?

What is the resultant velocity and Reynolds number at the bottom of the sprue if

the metal being cast is aluminum and has a viscosity of $0\mathrm{.}004N-\frac{s}{m^2}?$