Apply Newton's second law to the bucket's vertical motion:
F = ma
F = net force, m = mass of the bucket, a = acceleration of the bucket
Let us choose upward force to be positive and downward force to be negative. The net force F is the difference of the tension in the rope lifting the bucket and the weight of the bucket, i.e.:
F = T - W
F = net force, T = tension, W = weight
The weight of the bucket is given by:
W = mg
W = weight, m = mass, g = gravitational acceleration
Make some substitutions:
F = T - mg
T - mg = ma
Isolate T:
T = ma + mg
T = m(a+g)
Given values:
m = 5kg, a = 3m/s², g = 9.81m/s²
Plug in and solve for T:
T = 5(3+9.81)
T = 64.05N
