Dynamically equivalent system.

1. Mass dynamically equivalent system must satisfy the following 3 conditions. The sum of 2 masses must be equal to mass of rigid body. $m_1 + m_2 = m$

2. The centre of gravity of 2 masses must coincide with that of rigid body.

$m_1 L_1 = m_2 L_2$

3. The moment of inertia of two mass systems must be equal to moment of inertia of rigid body.

$IG_1 + IG_2 = IG$

$\frac{m_1 L_1}{L_2} = m_2$

$m_1 + \frac{m_1 L_1}{L_2} = m$

$m_1 L_2 + m_1 L_1 = m L_2$

$\frac{m_1 (L_1 + L_2)}{L_2} = m$

$M_1 = \frac{mL_2}{L_1 + L_2}$

$M_2 = \frac{ML_1}{L_1 + L_2}$

1. $IG_1 + IG_2 = IG$

$M_1 L_1^2 + m_2 L_2^2 = m k^2$

$\frac{mL_2(L_1)^2}{(L_1 + L_2)} + \frac{mL_1}{(L_1 + L_2)} (L_2)^2 = m k^2$

$\frac{L_2 (L_1)^2 + L_1 (L_2)^2}{(L_1 + L_2)} = k^2$

$k^2 = L_1 L_2$