In the figure shown a3 = 6m/s2 (downwards) and a2 = 4m/s2 (upwards). Find acceleration of 1.

Since the points 1,2,3 and are movable, so let their displacements are  x1 , x2 , x3 and x4 We observe that the length of the strings between 1 and 4 and 2 and 3 are constants.x1+x4=ℓ1;x2−x4+x3−x4=ℓ2Differentiating twice w.r.t time, we geta1+a4=0⇒a1=−a4→(1)a2−a4+a3−a4=0⇒a2+a3−2a4=0→(2)a2=−4ms−2,a3=6ms−2;−4+6−2a4=02a4=2⇒a4=1ms−2;  From(1), a1=−a4;a1=−1ms−2; a1=1ms−2 upwards