To solve for the $a_x$, the x-component of the acceleration, we need to first find the net force acting along the x-axis, and then apply Newton's second law, $F = ma$, to determine the acceleration.

Step 1: Breaking down the forces along the x-axis

Looking at the diagram, the forces that contribute to the x-component are:

1. The $5.0 \, \text{N}$ force, which is at an angle of $20^\circ$ to the x-axis.
2. The $1.0 \, \text{N}$ force acting along the positive x-axis.
3. The $3.0 \, \text{N}$ force acting along the negative x-axis.

We’ll break down the $5.0 \, \text{N}$ force into its x-component: $F_{5.0 \, \text{N}, x} = 5.0 \, \text{N} \cdot \cos(20^\circ)$ Using a calculator, $\cos(20^\circ) \approx 0.9397$, so: $F_{5.0 \, \text{N}, x} = 5.0 \times 0.9397 = 4.6985 \, \text{N}$

Now we have the x-components of the forces:

• $F_{1.0 \, \text{N}} = 1.0 \, \text{N}$ (positive x-direction)
• $F_{3.0 \, \text{N}} = -3.0 \, \text{N}$ (negative x-direction)
• $F_{5.0 \, \text{N}, x} = 4.6985 \, \text{N}$ (positive x-direction)

Step 2: Summing the forces along the x-axis

Now sum all the x-components of the forces: $F_{\text{net, x}} = 4.6985 \, \text{N} + 1.0 \, \text{N} - 3.0 \, \text{N} = 2.6985 \, \text{N}$

Step 3: Using Newton's second law

To find the acceleration, we use Newton’s second law: $a_x = \frac{F_{\text{net, x}}}{m}$ Given that the mass $m = 1.6 \, \text{kg}$: $a_x = \frac{2.6985 \, \text{N}}{1.6 \, \text{kg}} = 1.6866 \, \text{m/s}^2$

So the correct value of $a_x$ should be approximately: $a_x \approx 1.69 \, \text{m/s}^2$

Let me know if you'd like further clarification, or have any other questions.

Follow-up questions:

1. How do you calculate the y-component of the object's acceleration?
2. Can you derive the net force along the y-axis from the forces given in the diagram?
3. How would you calculate the resultant acceleration vector (magnitude and direction)?
4. How does changing the mass affect the acceleration for the same set of forces?
5. How would the problem change if the angle of the 5.0 N force were larger?

Tip:

Always make sure to carefully resolve vector components when dealing with forces at angles!