





You said:
Two forces, F⃗ 1
and F⃗ 2
 lift a bucket whose mass is m
 = 5.7 kg
 vertically upward. The magnitude of the gravitational acceleration is 9.80 m/s2
 . The magnitude of F1 is unknown.  The magnitude of F2
 = 93.0 N
 . Now you can apply Newton's 2nd Law in the y- axis to the find the bucket's acceleration ay.


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!

Two forces, F⃗ 1 and F⃗ 2 lift a bucket whose mass is m = 5.7 kg vertically upward. The magnitude of the gravitational acceleration is 9.80 m/s². The magnitude of F1 is unknown. The magnitude of F2 = 93.0 N. Now you can apply Newton's 2nd Law in the y-axis to find the bucket's acceleration ay.

Explore how to apply Newton's Second Law to find the acceleration of a bucket lifted by two forces. This detailed problem involves calculating net force, gravitational force, and solving for acceleration with given values of mass and forces. Suitable for high school physics students.

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