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# Answer to Question #108070 in Mechanics | Relativity for Valentine Agun

Question #108070
Four objects are held in position at the corners of a rectangle by light rods as shown in Figure P8.37. Find the moment of iner- tia of the system about (a)thex-axis,(b)they-axis, and (c) an axis through O and perpendicular to the page
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2020-04-06T09:03:56-0400

Solution

The moment of inertia of a mass m placed r meters from the axis of rotation is

"I=mr^2."

This is is an additive quantity. Therefore, the moment of inertia around x-axis will be

"I_x=m_1r_1^2+m_2r^2_2+m_3r^2_3+m_4r^2_4,"

where

"I_x=m_1r_1^2+m_2r^2_2+m_3r^2_3+m_4r^2_4,\\\\r_1=r_2=r_3=r_4=3\\text{ m}.\\\\\nI_x=(3.1+1.7+4.5+2.3)3^2=104\\text{ kg}\\cdot\\text{m}^2."

Consider rotation around y-axis:

"I_y=m_1r_1^2+m_2r^2_2+m_3r^2_3+m_4r^2_4,\\\\r_1=r_2=r_3=r_4=2\\text{ m}.\\\\\nI_y=(3.1+1.7+4.5+2.3)2^2=46.4\\text{ kg}\\cdot\\text{m}^2."

Now consider rotation around z-axis:

"I_z=m_1r_1^2+m_2r^2_2+m_3r^2_3+m_4r^2_4,\\\\r_1=r_2=r_3=r_4=\\sqrt{13}\\text{ m}.\\\\\nI_z=(3.1+1.7+4.5+2.3)\\sqrt{13^2}=151\\text{ kg}\\cdot\\text{m}^2."

Conclusion: the further, the higher.

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