Question1 Forces
19. Two ropes are attached to a tree, and forces of $\vec{F}_{1}=2.0 \hat{i}+4.0 \hat{j} \mathrm{~N}$ and $\vec{F}_{2}=3.0 \hat{i}+6.0 \hat{j} \mathrm{~N}$ are applied. The forces are coplanar (in the same plane). (a) What is the resultant (net force) of these two force vectors? (b) Find the magnitude and direction of this net force.

Studdy Solution
Determine the direction of the resultant force by calculating the angle $\theta$ with respect to the positive $\hat{i}$ axis using the tangent function.
$\tan \theta = \frac{\text{component along } \hat{j}}{\text{component along } \hat{i}} = \frac{10.0}{5.0}$
$\tan \theta = 2.0$
Calculate $\theta$ using the inverse tangent function:
$\theta = \tan^{-1}(2.0)$
$\theta \approx 63.43^\circ$
The direction of the resultant force is approximately $63.43^\circ$ above the positive $\hat{i}$ axis.