Questionwith David's rope. 5.2 Newton's First Law
22. Two forces of $\vec{F}_{1}=75.02(\hat{i}-\hat{j}) \mathrm{N}$ and $\vec{F}_{2}=\frac{150.0}{\sqrt{2}}(\hat{i}-\hat{j}) \mathrm{N}$ act on an object. Find the third force $\vec{F}_{3}$ that is needed to balance the first two forces.

Studdy Solution
Determine the third force needed to balance the resultant force.
The third force $\vec{F}_{3}$ must be equal in magnitude but opposite in direction to $\vec{F}_{\text{resultant}}$ to achieve equilibrium:
$\vec{F}_{3} = -\vec{F}_{\text{resultant}}$
$\vec{F}_{3} = - (181.09\hat{i} - 181.09\hat{j})$
$\vec{F}_{3} = -181.09\hat{i} + 181.09\hat{j}$
The third force needed is:
$\boxed{-181.09\hat{i} + 181.09\hat{j} \, \mathrm{N}}$