Question

In the reported figure, two bodies $A$ and $B$ of masses $200\, g$ and $800\, g$ are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be ______$rad / s$ when $k =20\, N / m$.

Answer 1

$\omega=\sqrt{\frac{k_{e q}}{\mu}}$
$\mu=$ reduced mass
springs are in series connection
$k_{e q}=\frac{k_{1} k_{2}}{k_{1}+k_{2}}$
$k_{e q}=\frac{k \times 4 k}{5 k}=\frac{4 k}{5}$
$k_{e q}=\frac{4 \times 20}{5} N / m=16\, N / m$
$\mu=\frac{m_{1} m_{2}}{m_{1}+m_{2}}=\frac{0.2 \times 0.8}{0.2+0.8}$
$=0.16\, kg$
$\omega=\sqrt{\frac{16}{0.16}}=\sqrt{100}=10$