As shown in the figure, two blocks of masses \(m_1\) and \(m_2\) are connected to a spring of force constant \(k\). The blocks are slightly displaced in opposite directions to \(x_1, x_2\) distances and released. If the system executes simple harmonic motion, then the frequency of oscillation of the system (\(\omega\)) is:
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In a system with two masses connected by a spring, the effective mass is given by the reduced mass formula. The frequency of oscillation depends on both the mass terms and the spring constant.
Step 1: Define the System's Dynamics
For a system of two masses connected by a spring with force constant \( k \), the frequency of oscillation can be determined using the concept of reduced mass.
Step 2: Use the Effective Mass Formula
The reduced mass of the system is given by:
\[
\mu = \frac{m_1 m_2}{m_1 + m_2}
\]
The angular frequency of oscillation is given by:
\[
\omega = \sqrt{\frac{k}{\mu}}
\]
Substituting \(\mu = \frac{m_1 m_2}{m_1 + m_2}\) into the equation:
\[
\omega = \sqrt{k \left( \frac{1}{m_1} + \frac{1}{m_2} \right)}
\]
