To find the mass of an object given the forces and acceleration, we can use Newton's Second Law of Motion:

$\vec{F}_{\text{net}} = m \cdot \vec{a}$

Where:

• $\vec{F}_{\text{net}}$ is the net force acting on the object (vector sum of $\vec{f}_1$ and $\vec{f}_2$),
• $m$ is the mass,
• $\vec{a}$ is the acceleration.

Step 1: Find the net force

The net force is the vector sum of $\vec{f}_1$ and $\vec{f}_2$: $\vec{F}_{\text{net}} = \vec{f}_1 + \vec{f}_2$

Given: $\vec{f}_1 = (-2.31 \, \text{N})\hat{i} + (2.79 \, \text{N})\hat{j}$ $\vec{f}_2 = (6.77 \, \text{N})\hat{i} + (-3.55 \, \text{N})\hat{j}$

Summing these vectors: $\vec{F}_{\text{net}} = \left( (-2.31) + 6.77 \right) \hat{i} + \left( 2.79 + (-3.55) \right) \hat{j}$ $\vec{F}_{\text{net}} = (4.46 \, \text{N})\hat{i} + (-0.76 \, \text{N})\hat{j}$

Step 2: Find the magnitude of the net force

The magnitude of the net force is found using the Pythagorean theorem: $|\vec{F}_{\text{net}}| = \sqrt{(4.46)^2 + (-0.76)^2}$

Step 3: Use Newton's Second Law to find the mass

From Newton's second law: $m = \frac{|\vec{F}_{\text{net}}|}{a}$ Where $a = 4.27 \, \text{m/s}^2$.

Let me calculate the mass for you.The mass of the object is approximately $1.06 \, \text{kg}$.

Would you like more details or have any further questions?

Here are some related questions you might find useful:

1. How do you find the resultant of two vectors?
2. What is Newton's Second Law and how is it applied in different situations?
3. How do vector components affect the overall magnitude of a vector?
4. How can acceleration vary if different forces are applied?
5. What are other methods for calculating force when mass and acceleration are known?

Tip: Always ensure the units of force and acceleration are consistent when applying Newton's second law to avoid calculation errors.