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The forces in the figure act on a 2 kg object. What is the value of the x and y components of the object’s acceleration?

To find the x and y components of the object's acceleration, we need to use Newton's Second Law, which states:

\[
\sum F = ma
\]

Where:
- \(\sum F\) is the net force acting on the object,
- \(m\) is the mass of the object (2 kg),
- \(a\) is the acceleration of the object.

The x and y components of the net force can be determined by breaking down each force vector into its respective components. Then we can use these components to find the acceleration in the x and y directions.

### Step 1: Break down the forces into components.

- **5.0 N force**: This force is at a \(20^\circ\) angle from the negative x-axis.
  - \(F_{5x} = 5.0 \, \text{N} \times \cos(20^\circ)\)
  - \(F_{5y} = 5.0 \, \text{N} \times \sin(20^\circ)\)

- **1.0 N force**: This force is along the positive x-axis.
  - \(F_{1x} = 1.0 \, \text{N}\)
  - \(F_{1y} = 0 \, \text{N}\) (no y-component)

- **3.0 N force**: This force is downward, meaning it acts in the negative y direction.
  - \(F_{3x} = 0 \, \text{N}\)
  - \(F_{3y} = -3.0 \, \text{N}\)

- **2.82 N force**: This force is along the positive y-axis.
  - \(F_{2.82x} = 0 \, \text{N}\)
  - \(F_{2.82y} = 2.82 \, \text{N}\)

### Step 2: Calculate the net forces in the x and y directions.

- For the x-direction:
  \[
  F_{net,x} = F_{5x} + F_{1x} + F_{3x} + F_{2.82x}
  \]
  
- For the y-direction:
  \[
  F_{net,y} = F_{5y} + F_{1y} + F_{3y} + F_{2.82y}
  \]

### Step 3: Use Newton’s Second Law to find the acceleration components.

- Acceleration in the x-direction:
  \[
  a_x = \frac{F_{net,x}}{m}
  \]

- Acceleration in the y-direction:
  \[
  a_y = \frac{F_{net,y}}{m}
  \]

I'll calculate the specific components now.The x and y components of the object's acceleration are:

- \( a_x = 2.85 \, \text{m/s}^2 \)
- \( a_y = 0.77 \, \text{m/s}^2 \)

Would you like more details or have any questions?

Here are some related questions you might find useful:
1. How do we calculate the resultant acceleration from the x and y components?
2. What is Newton's Second Law and how does it apply to objects in motion?
3. How would friction affect the forces and acceleration in this scenario?
4. How can trigonometric functions help in breaking down force vectors?
5. What would happen to the acceleration if the mass of the object were doubled?

**Tip**: When dealing with force problems, always break vectors into components before summing them to ensure accuracy!

This problem involves applying Newton's Second Law to determine the x and y components of acceleration for a 2 kg object subjected to multiple forces. Learn how to decompose forces using trigonometric functions and calculate the resultant acceleration. Suitable for high school physics students.

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