To solve this problem, we will apply Newton's Second Law in the vertical direction (y-axis) to find the bucket's acceleration $a_y$.

Step-by-Step Solution:

1. Identify the forces acting on the bucket:

   • Gravitational force: $F_g = m \cdot g = 5.7 \, \text{kg} \cdot 9.80 \, \text{m/s}^2 = 55.86 \, \text{N}$ (acting downward).
   • Force $F_1$ (acting upward), magnitude unknown.
   • Force $F_2 = 93.0 \, \text{N}$ (acting upward).

2. Net force in the y-axis: The net force $F_{\text{net}}$ acting on the bucket is the sum of the forces in the vertical direction: $F_{\text{net}} = F_1 + F_2 - F_g$ Where $F_1$ and $F_2$ are upward forces, and $F_g$ is the downward gravitational force.

3. Newton's Second Law: According to Newton’s second law: $F_{\text{net}} = m \cdot a_y$ So, $F_1 + F_2 - F_g = m \cdot a_y$ Substitute the known values: $F_1 + 93.0 \, \text{N} - 55.86 \, \text{N} = 5.7 \, \text{kg} \cdot a_y$ Simplifying: $F_1 + 37.14 \, \text{N} = 5.7 \, \text{kg} \cdot a_y$

4. Final equation: You now have a relationship between $F_1$ and the bucket's acceleration $a_y$. To find $a_y$, you need more information, such as the value of $F_1$. Without that, you can only express the acceleration in terms of $F_1$: $a_y = \frac{F_1 + 37.14 \, \text{N}}{5.7 \, \text{kg}}$

Would you like to continue with more details or ask additional questions? Here are some questions that can expand on this:

1. How does Newton’s second law apply when multiple forces are acting on a body?
2. How can we solve for $F_1$ if we know the bucket's acceleration?
3. How does gravity affect objects moving vertically upward?
4. What are the conditions for equilibrium in this system?
5. What happens if the acceleration is zero (i.e., the system is in equilibrium)?

Tip: Always ensure to consider the direction of forces carefully. Positive and negative signs are crucial in vector equations!