Question

At the given instant the four particles having masses and acceleration as shown in the fig lie at the vertices of a square. Acceleration of the mass of the system is:

• A. $\frac{1}{5}\left(\hat{i}+\hat{j}\right)$
• B. $\frac{1}{5}\left(\hat{j}-\hat{i}\right)$
• C. $\frac{1}{5}\left(\hat{i}-\hat{j}\right)$
• D. $-\frac{1}{5}\left(\hat{i}-\hat{j}\right)$

Answer 1

The correct option is C

$\frac{1}{5}\left(\hat{i}-\hat{j}\right)$

Step 1. Given Data

Masses of the four particles are $m_1,m_2,m_3,and{m}_4$

The values of the four particles are $m,2m,3m,and4m$respectively.

The acceleration of the four particles are $a_1,a_2,a_3,and{a}_4$

The values of acceleration of the four particles as per the diagram are $\left(-a\hat{i}\right),\left(a\hat{j}\right),\left(a\hat{i}\right)and\left(-a\hat{j}\right)$

Step 2. Find the acceleration of the system:

The acceleration of the system is given by the formula

$a_n=\frac{m_1{a}_1+m_2{a}_2+m_3{a}_3+m_4{a}_4}{m_1+m_2+m_3+m_4}$

$=\frac{m\left(-a\hat{i}\right)+2m\left(a\hat{j}\right)+3m\left(a\hat{i}\right)+4m\left(-a\hat{j}\right)}{m+2m+3m+4m}$

$=\frac{2\hat{i}-2\hat{j}}{10}$

$=\frac{1}{5}\left(\hat{i}-\hat{j}\right)$

Hence option (C) is correct