Question

A rod extending between x = 0 and x = 13.0 cm has uniform cross-sectional area A = 8.00 cm2. Its density increases steadily between its ends from 2.50 g/cm3 to 19.0 g/cm3. (a) Identify the constants B and C required in the expression rho = B + Cx to describe the variable density.

Answer 1

Answer:

The mass is  $m = 3.45408 kg$

Explanation:

From the question we are told that

    The extension  of the rod is $from , \ x_1 = 0 \to x_2 = 13.0$

     The area is  $A = 8.0 cm^2$

      The density increase as follows $from \ \rho_1 =2.5 g/cm^2 \to \rho_2= 19.0 g/cm^3$

    The equation  $\rho = B + Cx$

at  $x_1= 0$  $\rho_1 =2.5 g/cm^2$

So

      $2.5 = B + 0$

=>  $B =2.5$

    So at $x_2 = 13.0$ ,  $\rho_2= 19.0 g/cm^3$

So

            $19.0 = 2.5 + C(13)$

       =>   $C = 1.27$

Now  

       $m = 8 \int\limits^{13}_{0} {2.5 + 1.27x} \, dx$

      $m = 8 [{2.5 +\frac{ 1.27x^2}{2} } ]\left | 13} \atop {0}} \right.$

      $m = 8 [{2.5 +\frac{ 1.27(13)^2}{2} } ]$

      $m = 3454.08 g$

        $m = 3.45408 kg$