Question

In the HCI molecule, the separation between the nuclei of hydrogen and chlorine atoms is 1.27 $A^\circ $ .If the mass of a chlorine atom is 35.5 times that of a hydrogen atom, the center of mass of the HCI molecule is at a distance of
(A) $\dfrac{{35.5 \times 1.27}}{{36.5}}A$from the hydrogen atom
(B) $\dfrac{{35.5 \times 1.27}}{{36.5}}A$ from the chlorine atom
(C) $\dfrac{{1.27}}{{36.5}}A$from the hydrogen atom
(D) $\dfrac{{1.27}}{{36.5}}A$from the chlorine atom

Answer 1

Hint
The center of mass of a body is defined as a place where the whole mass of the body is supposed to be concentrated. In this question, we will take the mass of the hydrogen atom as m and the distance of hydrogen from the center of mass as x. Then the mass of the chlorine atom will become 35.5m. Use these values to find the center of mass of the system.

Complete step by step answer
As we know that the formula for finding the center of mass of a system of body is equal to:
$R\, = \,\dfrac{{\sum {{m_i}{r_i}} }}{{\sum {{m_i}} }}$
In this question, i is equal to 2. If we take the center of mass of the system to be origin, and the position of the other 2 bodies as x and $-\left( {1.27 - x} \right)$ , we get:
$\Rightarrow R\, = \,\dfrac{{mx - 35.5m(1.27 - x)}}{{m + 35.5m}} $
$\Rightarrow R\, = \,\dfrac{{x - 45.08 + 35.5x}}{{36.5m}} $
$\Rightarrow 45.08\, = \,36.5x $
$\Rightarrow x\, = \,1.23 $
Therefore, the center of mass is 1.23A away from the hydrogen atom and the correct answer is option (A).

Note
Center of mass has the unit of m instead of kg as intuitive it may sound. Also, it is an imaginary quantity and can lie on places where there is no mass at all. This case is realized with a uniform ring. The center of mass of the ring lies exactly at the geometrical center of the ring which is nothing but empty space.