(a) The free-body diagrams for each object are given below.
(b) Let $$a$$ represent the positive magnitude of the acceleration $$−a\hat{j}$$ of $$m_{1}$$, of the acceleration $$−a\hat{i}$$ of $$m_{2}$$, and of the acceleration $$+a\hat{j}$$ of $$m_{3}$$. Call $$T_{12}$$ the tension in the left cord and $$T_{23}$$ the tension in the cord on the right.
For $$m_{1}, \sum F_{y} = ma_{y} : +T_{12} − m_{1}g = −m_{1}a$$
For $$m_{2}, \sum F_{x} = ma_{x} : −T_{12} + \mu_{k}n + T_{23} = −m_{2}a$$
and $$\sum F_{y} = ma_{y}$$, giving $$n −m_{2}g = 0$$.
For $$m_{3}, \sum F_{y} = ma_{y}$$, giving $$T_{23} −m_{3}g = +m_{3}a$$.
We have three simultaneous equations:
$$−T_{12} + 39.2 N = (4.00 kg)a$$
$$+T_{12} − 0.350(9.80 N) − T_{23} = (1.00 kg)a$$
$$+T_{23} − 19.6 N = (2.00 kg)a$$
Add them up (this cancels out the tensions):
$$+39.2 N − 3.43 N − 19.6 N = (7.00 kg)a$$
$$a = 2.31 m/s^{2}$$, down for $$m_{1}$$, left for $$m_{2}$$, and up for $$m_{3}$$
(c) Now $$−T_{12} + 39.2 N = (4.00 kg)(2.31 m/s^{2} )$$
$$T_{12} = 30.0 N$$
and
$$T_{23} − 19.6 N = (2.00 kg)(2.31 m/s^{2} )$$
$$T_{23} = 24.2 N$$
(d) If the tabletop were smooth, friction disappears $$(\mu_{k} = 0)$$, and so the acceleration would become larger. For a larger acceleration, according to the equations above, the tensions change:
$$T_{12} = m_{1}g − m_{1}a\rightarrow T_{12}$$ decreases.
$$T_{23} = m_{3}g + m_{3}a\rightarrow T_{23}$$ increases.