The problem asks to calculate the moment of inertia of a system consisting of two masses, $m_1 = 2 \, \text{kg}$ and $m_2 = 3 \, \text{kg}$, connected by a massless rod, rotating around point $a$.

Given:

• $m_1 = 2 \, \text{kg}$
• $m_2 = 3 \, \text{kg}$
• Distance of $m_1$ from the axis of rotation at $a$ is $20 \, \text{cm} = 0.2 \, \text{m}$
• Distance of $m_2$ from the axis of rotation at $a$ is $30 \, \text{cm} = 0.3 \, \text{m}$

Formula for Moment of Inertia:

The moment of inertia $I$ for point masses is given by: $I = \sum m_i r_i^2$ where:

• $m_i$ is the mass,
• $r_i$ is the distance of the mass from the axis of rotation.

Calculations:

1. For $m_1$ (mass 1): $I_1 = m_1 \cdot r_1^2 = 2 \cdot (0.2)^2 = 2 \cdot 0.04 = 0.08 \, \text{kg} \cdot \text{m}^2$

2. For $m_2$ (mass 2): $I_2 = m_2 \cdot r_2^2 = 3 \cdot (0.3)^2 = 3 \cdot 0.09 = 0.27 \, \text{kg} \cdot \text{m}^2$

Total Moment of Inertia:

$I_{\text{total}} = I_1 + I_2 = 0.08 + 0.27 = 0.35 \, \text{kg} \cdot \text{m}^2$

So, the total moment of inertia of the system is $0.35 \, \text{kg} \cdot \text{m}^2$.

Do you need further details or have any questions?

Here are 5 related questions:

1. How would the moment of inertia change if the masses were swapped?
2. How does the distance from the axis of rotation affect the moment of inertia?
3. What happens to the moment of inertia if the system is rotated about a different point on the rod?
4. Can you derive the formula for moment of inertia for continuous bodies?
5. How is moment of inertia applied in real-world mechanical systems?

Tip: The moment of inertia depends heavily on both mass and the square of the distance from the axis of rotation, making distance a more dominant factor in the equation.