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:
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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.
