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

$\vec{F} = m \vec{a}$ where:

• $\vec{F}$ is the net force,
• $m$ is the mass of the object (1.2 kg), and
• $\vec{a}$ is the acceleration.

We will break the forces into their $x$- and $y$-components and use them to calculate the acceleration components.

1. Summing the forces in the $x$-direction:

From the diagram, the forces acting in the $x$-direction are:

• A force of $4.0 \, \text{N}$ to the left (negative $x$-direction),
• A force of $2.0 \, \text{N}$ to the right (positive $x$-direction).

The net force in the $x$-direction is: $F_x = -4.0 \, \text{N} + 2.0 \, \text{N} = -2.0 \, \text{N}$

2. Summing the forces in the $y$-direction:

In the $y$-direction, we have:

• A downward force of $3.0 \, \text{N}$,
• An upward force of $3.0 \, \text{N}$,
• A diagonal force of $3.0 \, \text{N}$ at an angle of 45 degrees above the negative $x$-axis.

The vertical component of the diagonal force (at 45 degrees) is: $F_{\text{diag}, y} = 3.0 \, \text{N} \times \sin(45^\circ) = 3.0 \, \text{N} \times \frac{\sqrt{2}}{2} \approx 2.12 \, \text{N}$

The net force in the $y$-direction is: $F_y = -3.0 \, \text{N} + 2.12 \, \text{N} = -0.88 \, \text{N}$

3. Calculating acceleration:

Now, applying Newton's second law $\vec{F} = m \vec{a}$ for each direction:

In the $x$-direction:

$a_x = \frac{F_x}{m} = \frac{-2.0 \, \text{N}}{1.2 \, \text{kg}} = -1.67 \, \text{m/s}^2$

In the $y$-direction:

$a_y = \frac{F_y}{m} = \frac{-0.88 \, \text{N}}{1.2 \, \text{kg}} \approx -0.73 \, \text{m/s}^2$

Final Answer:

• $a_x = -1.67 \, \text{m/s}^2$ (to the left),
• $a_y = -0.73 \, \text{m/s}^2$ (downward).

Let me know if you want further details or have any questions.

Related Questions:

1. How is the resultant acceleration of the object calculated?
2. What is the magnitude and direction of the net force acting on the object?
3. How does the angle of the diagonal force affect the components of acceleration?
4. What would happen to the acceleration if the mass of the object were doubled?
5. How do we resolve forces acting at an angle into their components?

Tip:

When dealing with forces at an angle, always break them into their $x$- and $y$-components using trigonometric functions like sine and cosine.