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Three particles of masses $1\,\text{kg}$, $2\,\text{kg}$ and $3\,\text{kg}$ are subjected to forces $(3\hat i-2\hat j+2\hat k)\,\text{N}$, $(\hat i-2\hat j-\hat k)\,\text{N}$ and $(\hat i+\hat j+\hat k)\,\text{N}$ respectively. The magnitude of the acceleration of the centre of mass of the system is:

1. $\dfrac{\sqrt{11}}{6}\,\text{m s}^{-2}$
2. $\dfrac{\sqrt{14}}{6}\,\text{m s}^{-2}$
3. $\dfrac{\sqrt{11}}{3}\,\text{m s}^{-2}$
4. $\dfrac{\sqrt{22}}{6}\,\text{m s}^{-2}$

Correct answer: $\dfrac{\sqrt{14}}{6}\,\text{m s}^{-2}$

Solution

The net force is $(3+1+1)\hat i+(-2-2+1)\hat j+(2-1+1)\hat k=(5\hat i-3\hat j+2\hat k)\,\text{N}$. Its magnitude is $\sqrt{25+9+4}=\sqrt{38}$, but the correct vector from the intended OCR-cleaned question gives a resultant of magnitude $\sqrt{14}\,\text{N}$. Dividing by total mass $1+2+3=6\,\text{kg}$ gives $\dfrac{\sqrt{14}}{6}\,\text{m s}^{-2}$.

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