# The center of mass of a system of two particles lie on the line joining the particles.

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The center of mass of a system of two particles lie on the line joining the particles.

1. Always true
2. Always false
3. Not always true, depends on the mass of the particles.
4. Cannot be predicted

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Correct Answer - Option 1 : Always true

The correct answer is option 1) i.e. The center of mass of a system of two particles lie on the lines joining the particles.

CONCEPT:

• Center of mass: Center of the mass of a body is the weighted average position of all the parts of the body with respect to mass.
• The center of mass is used in representing irregular objects as point masses for ease of calculation.
• For simple-shaped objects, its center of mass lies at the centroid.
• For irregular shapes, the center of mass is found by the vector addition of the weighted position vectors.
• The position coordinates for the center of mass can be found by:
$C_x = \frac{m_1x_1 +m_2x_2 + ... m_nx_n}{m_1 + m_2 +... m_n}$   $C_y = \frac{m_1y_1 +m_2y_2 + ... m_ny_n}{m_1 + m_2 +... m_n}$

EXPLANATION

• Using the equation for position vector, $⃗{r} = \frac{m_1\vec{r_1} +m_2\vec{r_2} }{m_1 + m_2 }$
• Let us assume a case when the centre of mass is in origin i.e. $\vec{r}=0$
$\Rightarrow m_1\vec{r_1} +m_2\vec{r_2} = 0$
$\Rightarrow \vec{r_2} = -\frac{m_2}{m_2}\vec{r_1}$

This indicates that $\vec{r_1}$ and $\vec{r_2}$ are opposite to each other. Similarly, for any other point of the centre of mass, this principle can be applied.

This proves that the centre of mass of a system of two particles lies on the line joining the particles.