# oscillations of blocks connected by a spring

Imagine two blocks of masses $m_1$ and $m_2$ joined together by a spring of spring constant $K$.

Now let the spring be stretched by a distance $X$ and then the system is released. suppose during the stretching the block of mass $m_1$ moves towards left by a distance $A$ and the block of mass $m_2$ by a distance B.

Now the centre of mass of the system at any instant will remain at rest. Now let it be present at a point P in between the two blocks. so can I depict the oscillations of the two blocks as two independent oscillations about the centre of mass?

$$m_1 \frac{{\rm d}^2\,a(t)}{{\rm d}t^2} = -F(t)$$ $$\mbox{-}m_2 \frac{{\rm d}^2\,b(t)}{{\rm d}t^2} = F(t)$$ $$F(t) = k \;\left( a(t)+b(t) \right)$$
Assume simple harmonic motion $a(t) = A \sin(\omega\,t)$, $b(t) = B \sin(\omega\,t)$ which leads to the frequency equation $$k\,\left(m_1+m_2\right) = m_1 m_2 \omega^2$$ and the amplitude equation $$A\, m_1 = B\, m_2$$ So now you can show that the center of gravity $P$ does not move as long as the amplitudes $A$ and $B$ obey the balance equation above. How? Well what is the equation for the center of gravity? $${\rm cg} = \frac{\mbox{-}B\,m_2+A\,m_1}{m_1+m_2} = 0$$