First slide

Location of centre on mass for multiple point masses

Question

Two particles whose masses are 30 kg and 10 kg and their position vectors are $- \hat{ı} - \hat{ȷ} - \hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$ respectively would have the centre of mass at position :

Moderate

A

$- (\hat{ı} + \hat{ȷ} + \hat{k}) / 2$
B

$\hat{ı} + \hat{ȷ} + \hat{k} / 2$
C

$- (\hat{ı} + \hat{ȷ} + \hat{k}) / 4$
D

$(\hat{ı} + \hat{ȷ} + \hat{k}) / 2$

Solution

$∵ {\vec{R}}_{cm} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2}}{m_{1} + m_{2}}$
$= \frac{10 (\hat{i} + \hat{j} + \hat{k}) + 30 (- \hat{i} - \hat{j} - \hat{k})}{10 + 30}$
$= \frac{10 \hat{i} + 10 \hat{j} + 10 \hat{k} - 30 \hat{i} - 30 \hat{j} - 30 \hat{k}}{40}$
$= \frac{- 20 \hat{i} - 20 \hat{j} - 20 \hat{k}}{40}$
${\vec{R}}_{c m} = \frac{- (\hat{i} + \hat{j} + \hat{k})}{2}$

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