Answer
a). $2.45m/s^{2}$,
b). $-1.96m/s^{2}$
Work Step by Step
Let tension in rope between $m_{1}$ and $m_{3}$ be $T_{1}$.
Let tension in rope between $m_{3}$ and $m_{2}$ be $T_{2}$.
$T_{1}-m_{1}g=m_{1}a$
$T_{1}=m_{1}(a+g)$
$m_{2}g-T_{2}=m_{2}a$
$T_{2}=m_{2}(g-a)$
$T_{2}-T_{1}=m_{3}a$
So, $m_{2}(g-a) -m_{1}(a+g)=m_{3}a$
$a(m_{1}+m_{2}+m_{3})=(m_{2}-m_{1})g$
or, $a=\frac{(m_{2}-m_{1})g}{(m_{1}+m_{2}+m_{3})}$
a). $m_{1}=0.25kg$, $m_{2}=0.5kg$, $m_{3}=0.25kg$, $a=2.45m/s^{2}$
b). $m_{1}=0.35kg$, $m_{2}=0.15kg$, $m_{3}=0.5kg$, $a=-1.96m/s^{2}$, in this case $m_{3}$ will accelerate towards the left.
