Question

Four objects are held in position at the corners of a rectangle by light rods as shown in the figure below. A. Find the moment of inertia of the system about the X axis. B. Find the moment of inertia of the system about the Y axis. C. Find the moment of inertia of the system about an axis through O and perpendicular to the page..

Answer 1

Given:

• m1 = 2.70 kg

,

• m2 = 2.30 kg

,

• m3 = 4.50 kg

,

• m4 = 1.90 kg

,

• Vertical distance = 6.00 m

,

• Horizontal distance = 4.00 m

Let's solve for the following:

• (a). Find the moment of inertia of the system about the X-axis.

To find the moment of inertia about the X-axis, apply the formula:

`I_x=\Sigma m(r_y)^2=r_y^2(m_1+m_2+m_3+m_4)`

Where:

ry = 6.00/2 = 3.00 m

Thus, we have:

`\begin{gathered} I_x=3.00^2(2.70+2.30+4.50+1.90) \\ \\ I_x=9(11.4) \\ \\ I_x=102.6\text{ kg.m}^2 \end{gathered}`

The moment of inertia about the x-axis is 102.6 kg.m².

• (b). Find the moment of inertia of the system about the Y-axis.

Apply the formula:

`I_y=r_x^2(m_1+m_2+m_3+m_4)`

Where:

rx = 4.00/2 = 2.00 m

We have:

`\begin{gathered} I_y=2.00^2(2.70+2.30+4.50+1.90) \\ \\ I_y=4.00(2.70+2.30+4.50+1.90) \\ \\ I_y=45.6\text{ kg.m}^2 \end{gathered}`

The moment of inertia about the y-axis is 45.6 kg.m².

• (c). Find the moment of inertia of the system about an axis through O and perpendicular to the page.

Apply the Perpendicular Axis Theorem.

We have:

`\begin{gathered} I_o=I_x+I_y \\ \\ I_o=102.6+45.6 \\ \\ I_o=148.2\text{ kg.m}^2 \end{gathered}`

ANSWER:

• (a). 102.6 kg.m².

,

• (b). 45.6 kg.m².

,

• (c). 148.2 kg.m².