To determine the [tex]$y$[/tex]-component of the total force acting on the block, we will decomposes the forces into their [tex]$y$[/tex]-components using trigonometric functions.
1. Understanding the problem:
- We have two forces:
- The first force [tex]\( F_1 = 115 \)[/tex] N at an angle [tex]\( \theta_1 = 34^\circ \)[/tex] from the horizontal.
- The second force [tex]\( F_2 = 213 \)[/tex] N at an angle [tex]\( \theta_2 = 29^\circ \)[/tex] from the horizontal.
- We need to find the resultant [tex]$y$[/tex]-component of these forces.
2. Identify the [tex]$y$[/tex]-component of each force:
- The [tex]$y$[/tex]-component of a force can be found using the sine of the angle because it gives the opposite side of the angle in a right triangle.
3. Calculate the [tex]$y$[/tex]-components:
- For the first force [tex]\( F_1 \)[/tex]:
[tex]\[
F_{1y} = F_1 \times \sin(\theta_1)
\][/tex]
Substituting the given values:
[tex]\[
F_{1y} = 115 \times \sin(34^\circ)
\][/tex]
This equals approximately [tex]\( 64.307 \)[/tex] N.
- For the second force [tex]\( F_2 \)[/tex]:
[tex]\[
F_{2y} = F_2 \times \sin(\theta_2)
\][/tex]
Substituting the given values:
[tex]\[
F_{2y} = 213 \times \sin(29^\circ)
\][/tex]
This equals approximately [tex]\( 103.264 \)[/tex] N.
4. Calculate the total [tex]$y$[/tex]-component:
- The total [tex]$y$[/tex]-component is the sum of the individual [tex]$y$[/tex]-components:
[tex]\[
F_y = F_{1y} + F_{2y}
\][/tex]
Substituting the calculated components:
[tex]\[
F_y = 64.307 + 103.264
\][/tex]
This equals approximately [tex]\( 167.572 \)[/tex] N.
Therefore, the [tex]$y$[/tex]-component of the total force acting on the block is:
[tex]\[
\overrightarrow{F_y} = 167.572 \, \text{N}
\][/tex]
