Question

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

Answer 1

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

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

Consider rotation around y-axis:

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

Now consider rotation around z-axis:

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

Conclusion: the further, the higher.