Oscillation Of Two Particle System - Practice Questions & MCQ

Two blocks of masses $m_1$ and $m_2$ are connected with a spring of natural length I and spring constant k . The system is lying on a
frictionless horizontal surface. Initially, the spring is compressed by a distance  as shown in below Figure.

If we release these blocks from the compressed position, then they will oscillate and will perform SHM about their equilibrium position.
- The time period of the blocks-

In this case, the reduced mass $\mathrm{m}_{\mathrm{r}}$ is given by $\frac{1}{m_r}=\frac{1}{m_1}+\frac{1}{m_2}$
and

$
T=2 \pi \sqrt{\frac{m_r}{k}}
$

Or
- The amplitude of the blocks- Let the amplitude of the blocks as $\mathrm{A}_1$ and $\mathrm{A}_2$
then $m_1 A_1=m_2 A_2$
(As net external force is zero and initially the centre of mass was at rest

$
\text { so } \Delta x_{c m}=0 \text { ) }
$

By energy conservation,

$
\begin{aligned}
& \frac{1}{2} k\left(A_1+A_2\right)^2=\frac{1}{2} k x^2 \\
& A_1+A_2=x_0 \quad \text { or, } \quad A_1+\frac{m_1}{m_2} A_1=x_0 \\
& A_1=\frac{m_2 x_0}{m_1+m_2}
\end{aligned}
$

Similarly, $A_2=\frac{m_1 x_0}{m_1+m_2}$