Answer
$T=50.42N$
Work Step by Step
Remember that the net force is equal to mass times acceleration, or $$F_{net}=ma$$ The weight of the bucket pulls the bucket downward. This weight is equal to $mg$, which is $(4.35kg)(9.80m/s^2)=42.6N$. This force is acting in the negative direction, so $F_w=-42.6N$. The acceleration of the object is equal to $1.78m/s^2$, so the net force must be equal to $(4.35kg)(1.78m/s^2)=7.743N$. By drawing a free-body diagram, it can be concluded that $$T-42.6N=7.74N$$ where $T$ is the tension. This was concluded since all forces add up to a net force of 7.74N in this question. Solving for tension yields a value of $T=50.42N$.
