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Center of Mass question

2005 · Shift 0 · Q182
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Center of Mass question

2005 · Shift 0 · Q182

JEE MainPhysicsCenter of MassMCQ+4 / −1
A body AAA of mass MMM while falling vertically downloads under gravity breaks into two-parts; a body BBB of mass 13M{1 \over 3}M31​M and a body CCC of mass 23M.{2 \over 3}M.32​M. The center of mass of bodies BBB and CCC taken together shifts compared to that of bodies BBB and CCC taken together shifts compared to that of body AAA towards
  1. A
    does not shift
  2. B
    depends on height of breaking
  3. C
    body BBB
  4. D
    body CCC
View written solutionFree

Correct answer: A

  1. Key principle: motion of the center of mass

When a body breaks into parts due to internal forces, the position of the center of mass (COM) of the system is not affected by how the mass is distributed internally.

The only external force acting on the system of fragments BBB and CCC is gravity.

So, immediately after breaking, the center of mass of the two fragments must continue to move exactly as the original body AAA would have moved.

  1. Before breaking

The body AAA of mass MMM is falling vertically downward.

Its center of mass is simply at the position of the body itself.

  1. After breaking

The body breaks into:

  • BBB of mass 13M\dfrac{1}{3}M31​M
  • CCC of mass 23M\dfrac{2}{3}M32​M

The center of mass of the two-body system is

R⃗CM=mBr⃗B+mCr⃗CmB+mC\vec R_{\text{CM}}=\frac{m_B\vec r_B+m_C\vec r_C}{m_B+m_C}RCM​=mB​+mC​mB​rB​+mC​rC​​

But at the instant of breaking, both fragments originate from the same point where body AAA was present. Hence the COM at that instant remains at the same position as that of AAA.

Also, subsequently, since only gravity acts externally, the COM continues along the same trajectory that body AAA would have followed.

  1. Conclusion

Therefore, the center of mass of bodies BBB and CCC taken together does not shift compared to that of body AAA.

So the correct option is:

A: does not shift\boxed{\text{A: does not shift}}A: does not shift​
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