Question

Two bodies of mass \(1 \mathrm{~kg}\) and \(3 \mathrm{~kg}\) have position vector \((\mathrm{i} \wedge+2 \mathrm{j} \wedge+\mathrm{k} \wedge)\) and $(-3 \mathrm{i}-2 \mathrm{j}+\mathrm{k})$ respectively the centre of mass of this system has a position vector ..... \(\\{\mathrm{A}\\}-2 \mathrm{i} \wedge+2 \mathrm{k} \wedge\) \(\\{B\\}-2 i \wedge-j \wedge+k \wedge\) \(\\{C\\} 2 i \wedge-j \wedge-k \wedge\) \(\\{\mathrm{D}\\}-\mathrm{i} \wedge+\mathrm{j} \wedge+\mathrm{k} \wedge\)

Short answer

The position vector of the center of mass of this system is \(-2 i \wedge-j \wedge+k \wedge\).

Step-by-step solution

Write down the given information

We have: - Masses: \(m_1 = 1 \mathrm{~kg}\) and \(m_2 = 3 \mathrm{~kg}\) - Position vectors: \(\vec{r}_1 = (\mathrm{i} \wedge+2 \mathrm{j} \wedge+\mathrm{k} \wedge)\) and \(\vec{r}_2 = (-3\mathrm{i}-2 \mathrm{j}+\mathrm{k})\)

Calculate the center of mass position vector

Using the formula for the center of mass position vector, we have: $$\vec{R}_{\mathrm{cm}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2}$$ Substitute the given values into the formula: $$\vec{R}_{\mathrm{cm}} = \frac{1(\mathrm{i} \wedge+2 \mathrm{j} \wedge+\mathrm{k} \wedge) + 3(-3\mathrm{i}-2 \mathrm{j}+\mathrm{k})}{1+3}$$

Simplify the position vector

Now, perform the necessary operations: $$\vec{R}_{\mathrm{cm}} = \frac{1\,\mathrm{i} + 2\,\mathrm{j} + \mathrm{k} - 9\,\mathrm{i} - 6\,\mathrm{j} + 3\,\mathrm{k}}{4}$$ Combine the like terms: $$\vec{R}_{\mathrm{cm}} = \frac{-8\,\mathrm{i} - 4\,\mathrm{j} + 4\,\mathrm{k}}{4}$$ Divide each term by the denominator (4): $$\vec{R}_{\mathrm{cm}} = -2\,\mathrm{i} - \mathrm{j} + \mathrm{k}$$

Compare to the given options

We now compare our result to the given options: A) \(-2 \mathrm{i} \wedge+2 \mathrm{k} \wedge\) B) \(-2 i \wedge-j \wedge+k \wedge\) C) \(2 i \wedge-j \wedge-k \wedge\) D) \(-\mathrm{i} \wedge+\mathrm{j} \wedge+\mathrm{k} \wedge\) Our result matches option B: $$\vec{R}_{\mathrm{cm}} = -2 i \wedge-j \wedge+k \wedge$$ So, the position vector of the center of mass of this system is given by option B.