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# Calculating the Moment of Inertia of a System of Three Masses Two balls with masses 2 kg, 2 kg, and 3 kg are arranged at the corners of an equilateral triangle. The side length of the triangle is *l*. The system is viewed in a plane, and an axis passes through the centroid (O) of the triangle. Th...

# Calculating the Moment of Inertia of a System of Three Masses Two balls with masses 2 kg, 2 kg, and 3 kg are arranged at the corners of an equilateral triangle. The side length of the triangle is *l*. The system is viewed in a plane, and an axis passes through the centroid (O) of the triangle. The problem asks to calculate the moment of inertia of the system about this axis. **Given:** * Mass of the first ball ($m_1$) = 2 kg * Mass of the second ball ($m_2$) = 2 kg * Mass of the third ball ($m_3$) = 3 kg * Side length of the equilateral triangle ($l$) is a given value. **Diagram Description:** A sketch of an equilateral triangle is shown, with the masses positioned at its corners. A third mass (m3) is positioned at the apex. The centroid (O) of the triangle has lines from each corner connecting to the centroid. **Calculations:** The provided solution shows various steps calculating the distances from the centroid to each mass. It is derived that: * $AO = \frac{2}{3} AM$ * $CO = \frac{2}{3} CP$ * $BO = \frac{2}{3} BN$ Where AM, CP, and BN are segments from the centroid to the midpoints of each side, respectively. This means the distance from the centroid to either midpoint is equal to $\frac{1}{\sqrt 3}$ times the side length. Therefore, $AM = CP = BN = \frac{l}{\sqrt 3}$. Next, distances from the centroid are substituted into the equation: $I = m_1 u_1^2 + m_2 u_2^2 + m_3 u_3^2$ $I = m_1( \frac{1}{\sqrt{3}} )^2 l/3 +m_2( \frac{1}{\sqrt{3}} )^2 l/3 +m_3( \frac{1}{\sqrt{3}} )^2 l/3$ $I = 2(\frac{1}{3}) + 2(\frac{1}{3}) + 3(\frac{1}{3})$ The solution concludes by simplifying the equation to arrive at the final moment of inertia formula. Therefore, the final formula for the moment of inertia is shown and calculated in the given image.