Chapter 8 - Section 8.3 - Applications to Physics and Engineering. - 8.3 Exercises - Page 585: 24

$$\eqalign{ & {M_x} = - 2{\text{ and }}{M_y} = 15 \cr & \left( {\overline x ,\overline y } \right) = \left( {0.9375, - 0.125} \right) \cr} $$

$$\eqalign{ & {m_1} = 4,{\text{ }}{m_2} = 3,{\text{ }}{m_3} = 6,{\text{ }}{m_4} = 3 \cr & {P_1}\left( {6,1} \right),{\text{ }}{P_2}\left( {3, - 1} \right),{\text{ }}{P_3}\left( { - 2,2} \right),{\text{ }}{P_4}\left( { - 2, - 5} \right) \cr & \cr & {\text{The total mass of the system is}} \cr & m = {m_1} + {m_2} + {m_3} + {m_4} \cr & m = 4 + 3 + 6 + 3 \cr & m = 16 \cr & \cr & {\text{*The moment about the }}y{\text{ - axis is }} \cr & {M_y} = {m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3} + {m_4}{x_4} \cr & {M_y} = \left( 4 \right)\left( 6 \right) + \left( 3 \right)\left( 3 \right) + \left( 6 \right)\left( { - 2} \right) + \left( 3 \right)\left( { - 2} \right) \cr & {M_y} = 24 + 9 - 12 - 6 \cr & {M_y} = 15 \cr & \cr & {\text{*The moment about the }}x{\text{ - axis is }} \cr & {M_x} = {m_1}{y_1} + {m_2}{y_2} + {m_3}{y_3} \cr & {M_x} = \left( 4 \right)\left( 1 \right) + \left( 3 \right)\left( { - 1} \right) + \left( 6 \right)\left( 2 \right) + \left( 3 \right)\left( { - 5} \right) \cr & {M_x} = 4 - 3 + 12 - 15 \cr & {M_x} = - 2 \cr & \cr & {\text{The moments are:}} \cr & {M_x} = - 2{\text{ and }}{M_y} = 15 \cr & \cr & {\text{The center of mass of the system is:}} \cr & \overline x = \frac{{{M_y}}}{m} = \frac{{15}}{{16}} = 0.9375 \cr & \overline y = \frac{{{M_x}}}{m} = \frac{{ - 2}}{{16}} = - 0.125 \cr & \cr & \left( {\overline x ,\overline y } \right) = \left( {0.9375, - 0.125} \right) \cr} $$