To determine the [tex]\( y \)[/tex]-component of the total force acting on the block, let's break down the problem step by step:
1. Identify the forces and their angles:
- The first force [tex]\( F_1 \)[/tex] has a magnitude of 78.5 N and is directed at an angle of [tex]\( 0^\circ \)[/tex] from the positive x-axis.
- The second force [tex]\( F_2 \)[/tex] has a magnitude of 96.5 N and is directed at an angle of [tex]\( 273^\circ \)[/tex] from the positive x-axis.
2. Calculate the [tex]\( y \)[/tex]-component of each force:
The [tex]\( y \)[/tex]-component of a force can be found using the formula:
[tex]\[
F_y = F \sin(\theta)
\][/tex]
- For the first force [tex]\( F_1 \)[/tex] at [tex]\( 0^\circ \)[/tex]:
[tex]\[
F_{1y} = 78.5 \sin(0^\circ)
\][/tex]
Since [tex]\( \sin(0^\circ) = 0 \)[/tex], the [tex]\( y \)[/tex]-component of the first force is:
[tex]\[
F_{1y} = 78.5 \times 0 = 0.0 \, \text{N}
\][/tex]
- For the second force [tex]\( F_2 \)[/tex] at [tex]\( 273^\circ \)[/tex]:
[tex]\[
F_{2y} = 96.5 \sin(273^\circ)
\][/tex]
The sine of [tex]\( 273^\circ \)[/tex] is a negative value because [tex]\( 273^\circ \)[/tex] is in the fourth quadrant where the sine function is negative. More specifically:
[tex]\[
\sin(273^\circ) \approx \sin(360^\circ - 87^\circ) = -\sin(87^\circ) \approx -0.998
\][/tex]
Therefore:
[tex]\[
F_{2y} = 96.5 \times -0.998 = -96.36775010381638 \, \text{N}
\][/tex]
3. Sum the [tex]\( y \)[/tex]-components of both forces:
[tex]\[
\text{Total } F_y = F_{1y} + F_{2y} = 0.0 + (-96.36775010381638)
\][/tex]
So, the [tex]\( y \)[/tex]-component of the total force is:
[tex]\[
\overrightarrow{F_y} \approx -96.36775010381638 \, \text{N}
\][/tex]
Thus, the [tex]\( y \)[/tex]-component of the total force acting on the block is:
[tex]\[
\boxed{-96.36775010381638 \, \text{N}}
\][/tex]
