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Let there be two blocks $m_1$ and $m_2$ both are attched with a spring and have a velocity $v_1$ and $v_2$ respectively . They are attached to same spring.

Now I have some doubts related to it.

  1. With respect to centre of mass, how these blocks are doing SHM.

  2. And in ground frame can we use energy conservation like this $$(1/2)m_1v_1^2+(1/2)m_2v_2^2=(1/2)kx^2$$ Where $x$ is maximum compression in spring. Where k is spring constant

  3. Can we use concept of reduce mass in such situations

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  • $\begingroup$ Have you looked at the mall you other questions on this topic, e.g. this one? What is still unclear? $\endgroup$
    – Floris
    Commented Jan 31, 2017 at 12:01

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Regarding point 1: if you consider the center of mass stationary (observe the two blocks in their center of mass frame), then the two blocks will move towards each other and away again. The velocity of one ($m_1$) will always be $\frac{m_2}{m_1}$ of the other (conservation of momentum). Considering the c.o.m. as the origin, that origin will be fixed and each mass will move as though it only "sees" the bit of spring on its side of the c.o.m. (dashed line = location of center of mass):

enter image description here

It follows that you can use reduced mass (although I prefer, from the visual above, to use the normal mass and scale the $k$ of the spring. Same result, mathematically). And if you look for conservation of energy, at any moment the sum of kinetic energies of the two blocks plus the elastic energy stored will be constant. However, it's not clear that your expression in (2) would be correct - there is no reason that the velocity of the two blocks would be a maximum at the same time (except in the c.o.m. frame).

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  • $\begingroup$ Here how you have proved they are having SHM $\endgroup$
    – user123733
    Commented Jan 31, 2017 at 3:15
  • $\begingroup$ In c.o.m frame why their velocity will we maximum at the same time $\endgroup$
    – user123733
    Commented Jan 31, 2017 at 3:16
  • $\begingroup$ If the second diagram is equal to the first for the RHS then it is a mass on a spring and it will exhibit SHM. Since the have the same frequency (otherwise the com would not be stationary) they must be in antiphase at all times. So they reach max velocity at the same time... $\endgroup$
    – Floris
    Commented Jan 31, 2017 at 3:19
  • $\begingroup$ Its not necessary that the c.o.m is at rest . It depends on the data taken $\endgroup$
    – user123733
    Commented Jan 31, 2017 at 3:22
  • $\begingroup$ Eh - yes. In the center of mass frame that is exactly what that means. No external forces. $\endgroup$
    – Floris
    Commented Jan 31, 2017 at 3:23

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