Chapter 8 - Section 8.3 - Applications to Physics and Engineering. - 8.3 Exercises - Page 585: 23

$$\eqalign{ & {M_x} = 23{\text{ and }}{M_y} = - 20 \cr & \left( {\overline x ,\overline y } \right) = \left( { - 1,1.15} \right) \cr} $$

$$\eqalign{ & {m_1} = 5,{\text{ }}{m_2} = 8,{\text{ }}{m_3} = 7 \cr & {P_1}\left( {3,1} \right),{\text{ }}{P_2}\left( {0,4} \right),{\text{ }}{P_3}\left( { - 5, - 2} \right) \cr & {\text{The total mass of the system is}} \cr & m = {m_1} + {m_2} + {m_3} \cr & m = 5 + 8 + 7 \cr & m = 20 \cr & \cr & {\text{*The moment about the }}y{\text{ - axis is }} \cr & {M_y} = {m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3} \cr & {M_y} = \left( 5 \right)\left( 3 \right) + \left( 8 \right)\left( 0 \right) + \left( 7 \right)\left( { - 5} \right) \cr & {M_y} = 15 + 0 - 35 \cr & {M_y} = - 20 \cr & \cr & {\text{*The moment about the }}x{\text{ - axis is }} \cr & {M_x} = {m_1}{y_1} + {m_2}{y_2} + {m_3}{y_3} \cr & {M_x} = \left( 5 \right)\left( 1 \right) + \left( 8 \right)\left( 4 \right) + \left( 7 \right)\left( { - 2} \right) \cr & {M_x} = 5 + 32 - 14 \cr & {M_x} = 23 \cr & \cr & {\text{The moments are:}} \cr & {M_x} = 23{\text{ and }}{M_y} = - 20 \cr & \cr & {\text{The center of mass of the system is:}} \cr & \overline x = \frac{{{M_y}}}{m} = \frac{{ - 20}}{{20}} = - 1 \cr & \overline y = \frac{{{M_x}}}{m} = \frac{{23}}{{20}} = 1.15 \cr & \cr & \left( {\overline x ,\overline y } \right) = \left( { - 1,1.15} \right) \cr} $$