Question

Two bodies of mass \( 1 \) kg and \( 3 \) kg have position vectors \( \hat{i} + 2\hat{j} + \hat{k} \) and \( -3\hat{i} - 2\hat{j} + \hat{k} \) respectively. The magnitude of the position vector of the center of mass of this system will be similar to the magnitude of which vector?

• A. \( \hat{i} - 2\hat{j} + \hat{k} \)
• B. \( -3\hat{i} - 2\hat{j} + \hat{k} \)
• C. \( -2\hat{i} + 2\hat{k} \)
• D. \( -2\hat{i} - \hat{j} + 2\hat{k} \)

Hint:

The center of mass position is given by \( \vec{r}_{{com}} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \). To find the correct vector, compare magnitudes.

Answer 1

The Correct Option is A

Step 1: {Formula for center of mass} 
\[ \vec{r}_{{com}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2} \] 
Step 2: {Substituting values} 
\[ \vec{r}_{{com}} = \frac{(1)(\hat{i} + 2\hat{j} + \hat{k}) + (3)(-3\hat{i} - 2\hat{j} + \hat{k})}{1 + 3} \] \[ = \frac{\hat{i} + 2\hat{j} + \hat{k} - 9\hat{i} - 6\hat{j} + 3\hat{k}}{4} \] \[ = \frac{-8\hat{i} - 4\hat{j} + 4\hat{k}}{4} \] \[ = -2\hat{i} - \hat{j} + \hat{k} \] 
Step 3: {Find the magnitude} 
\[ |\vec{r}_{{com}}| = \sqrt{(-2)^2 + (-1)^2 + (1)^2} \] \[ = \sqrt{4 + 1 + 1} = \sqrt{6} \] The only vector with the same magnitude is \( \hat{i} - 2\hat{j} + \hat{k} \).