The question is asking for the moment of inertia of a mass system arranged in a square layout with masses $m_1$, $m_2$, $m_3$, and $m_4$ placed at each corner. The rotational axis runs vertically through the center of the square, as indicated by the image. Let's solve this by applying the formula for the moment of inertia of point masses:

Solution Outline

For a point mass, the moment of inertia $I$ about an axis at a distance $r$ is given by: $I = m \cdot r^2$

In this case:

• $m_1$ and $m_2$ are at a horizontal distance of $s_A$ from the rotational axis.
• $m_3$ and $m_4$ are at a horizontal distance of $s_B$ from the rotational axis.

Thus, the total moment of inertia $I_{\text{total}}$ will be the sum of the individual moments of inertia of each mass:

$I_{\text{total}} = m_1 \cdot s_A^2 + m_2 \cdot s_A^2 + m_3 \cdot s_B^2 + m_4 \cdot s_B^2$

This can be rearranged as: $I_{\text{total}} = (m_1 + m_2) \cdot s_A^2 + (m_3 + m_4) \cdot s_B^2$

Answer Selection

Based on the answer choices: The correct answer matches the format: $(m_1 + m_2) \cdot s_A^2 + (m_3 + m_4) \cdot s_B^2$

This is the last option provided in the choices.

Would you like further details or have any questions?

Here are five related questions for additional practice:

1. How does the distance of each mass from the rotational axis affect the moment of inertia?
2. What would the moment of inertia be if all four masses were the same?
3. How would the moment of inertia change if the rotational axis were shifted?
4. Can this setup be extended to a three-dimensional arrangement?
5. What are other common shapes of mass distributions in physics?

Tip: Remember, the moment of inertia is greatly influenced by the square of the distance from the axis—small changes in distance have a big impact.