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. C.)

Explanation / Answer

This is a square with side = 6m

with the rotation point being the lower left corner and motion in the x-y plane

First, find the moment of inertia, which is generally given by the formula:

I = m * r^2

r = the distance from the rotation axis.

Moments of inertia are additive, so find the moment of inertia for each particle that's moving about the z-axis and add them together:

I7 = 7kg * (6m)^2

= 42 Kg m^2

I2 = 2kg * 72

(by Pythagoras: 6^2 + 6^2)

= 144 kg m^2

I7 = 7Kg * (6m)^2

= 252 kg m^2

Itotz = 42 + 144 + 144

= 330 kg m^2 [ moment of inertia about z]

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