# 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?

The two oscillations will not be independent as they will share frequency and phase. You can start from Newton's equations of motion. $$m_1 \frac{{\rm d}^2\,a(t)}{{\rm d}t^2} = -F(t)$$ $$\mbox{-}m_2 \frac{{\rm d}^2\,b(t)}{{\rm d}t^2} = F(t)$$ $$F(t) = k \;\left( a(t)+b(t) \right)$$

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? $${\rm cg} = \frac{\mbox{-}B\,m_2+A\,m_1}{m_1+m_2} = 0$$

• Did you make that image just for this problem? – Alan Rominger Aug 4 '11 at 14:24
• yes - and it is pretty cruddy too. In fact, 80% of solving a problem is framing the problem and making a nice sketch of all relevant information. – ja72 Aug 4 '11 at 16:08
• I found it to be rather impressive. This is a high quality answer overall IMO. – Alan Rominger Aug 4 '11 at 16:21

Traditionally, the "mass on a spring" is analyzed with one end of the spring held fixed. Your problem is equivalent to two such systems back-to-back. You have identified the center of gravity as a fixed point; so you can literally fix it and then cut the problem in two. Each half of the system then behaves like a traditional "mass-on-a-spring".