Four particles with masses 7 kg, 2 kg, 7 kg, and 2 kg are connected by rigid rod

Question

Four particles with masses 7 kg, 2 kg, 7 kg, and 2 kg are connected by rigid rods of neg- ligible mass as shown. The origin is centered on the mass in the lower left corner. The rectangle is 6 m wide and 4 m long.a.) If the system rotates in the xy plane about the z axis (origin, O) with an angular speed of 5 rad/s, calculate the moment of inertia of the system about the z axis.b.) Find the moment of inertia of the four-particle system about an axis that is perpendicular to the plane of the configuration and passing through the center of mass of the system. c.) For the four-particle system find the mo- ment of inertia about the y-axis, which passes through the two masses on the left-hand side of the system.

Explanation / Answer

each particle is positioned at r=v((5.4/2)^2 +(4/2)^2) from z-axis; kinetic energy of the system is E=E1+E2+E3+E4, where E1=0.5m1*v^2, m1=m2=2kg, m3=3kg, m4=4kg, v=w*r is their linear speed, w=5.5rad/s; thus we get E=0.5*m*(w*r)^2, where m= m1+m2+m3+m4 =11kg; E=0.5*11*5.5^2 *((5.4/2)^2 +(4/2)^2) =1878.37375 J; Also E=0.5*J*w^2, where J is moment of inertia of the system; Thus 0.5*m*(w*r)^2 = 0.5*J*w^2, hence J=m*r^2 = 11*((5.4/2)^2 +(4/2)^2) =124.19 kg*m^2;

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