Question

For the arrangement shown in the figure, $m_1=6\text{kg}$ and $m_2=4\text{kg}$. If the string is light and inextensible and the pulley and surfaces are frictionless, find the magnitude of the acceleration of the center of mass.
(Take $g=10m/s^2$)

• A. $4{\text{m/s}}^2$
• B. $10{\text{m/s}}^2$
• C. $6{\text{m/s}}^2$
• D. $2.88{\text{m/s}}^2$

Answer 1

The correct option is D $2.88{\text{m/s}}^2$

FBD of Blocks


For $6\text{kg}$ block
$T=6a\left(i\right)$

For 4 kg block ,
$\text{m}g-T=\text{m}a$
$4g-T=4a$
$4g-4a=T\left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$,
$4g-4a=6a$
$\Rightarrow 4g=10a$
$\Rightarrow a=4{\text{m/s}}^2$

Therefore, acceleartion of $6\text{kg}$ block is $4{\text{m/s}}^2$ along $(+x)$- axis while the acceleration of $4\text{kg}$ block is $4{\text{m/s}}^2$ along $(-y)$ - axis.

$a_{com}=\frac{m_1{a}_1+m_2{a}_2}{m_1+m_2}=\frac{6\times 4\hat{i}-4\times 4\hat{j}}{10}=2.4\hat{i}-1.6\hat{j}$

$|a_{com}|=\sqrt{{2.4}^2+(-1.6{)}^2}=2.88{\text{m/s}}^2$