Center of Mass, Momentum & Collision

JEE Main level, single correct, medium

41 of 130

Question 41 of 130

Two bodies of mass 1kg1kg and 3kg3kg have position vectors i^+2j^+k^\hat{i}+2\hat{j}+\hat{k} and −3i^−2j^+k^-3\hat{i}-2\hat{j}+\hat{k} respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

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Answer: (A) i^−2j^+k^\hat{i}-2\hat{j}+\hat{k}

As we know, centre of mass is given by, r⃗com=∑mir⃗i∑mi{\vec{r}}_{com}=\dfrac{\sum {m}_{i}{\vec{r}}_{i}}{\sum {m}_{i}}
r⃗com=m(ı^+2ȷ^+k^)+3m(−3ı^−2ȷ^+k^)m+3m{\vec{r}}_{com}=\dfrac{m\left(\hat{ı}+2\hat{ȷ}+\hat{k}\right)+3m\left(-3\hat{ı}-2\hat{ȷ}+\hat{k}\right)}{m+3m}
r⃗com=−8mı^−4mȷ^+4mk^4m=−2ı^−ȷ^+k^{\vec{r}}_{com}=\dfrac{-8m\hat{ı}-4m\hat{ȷ}+4m\hat{k}}{4m}=-2\hat{ı}-\hat{ȷ}+\hat{k}
The magnitude of the position vector of the centre of mass is 22+11+12=6\sqrt{{2}^{2}+{1}^{1}+{1}^{2}}=\sqrt{6}.
It is similar to the magnitude of the option A.

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