To determine the net force in the [tex]\( y \)[/tex]-direction when a box is pushed down at an angle of 32 degrees on a rough surface, let's break down the forces involved:
1. Normal Force ( [tex]\( F_N \)[/tex] ):
The force exerted by the surface perpendicular to the box.
2. Gravitational Force ( [tex]\( F_g \)[/tex] ):
The force due to gravity acting on the box vertically downward.
3. Push Force ( [tex]\( F_p \)[/tex] ):
The force applied at an angle of 32 degrees to the horizontal. This force has both [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-components.
To analyze the forces in the [tex]\( y \)[/tex]-direction:
- The gravitational force [tex]\( F_g \)[/tex] acts downward.
- The normal force [tex]\( F_N \)[/tex] acts upward, perpendicular to the surface.
- The vertical component of the push force [tex]\( F_p \sin(32) \)[/tex] acts downward.
Therefore, the net force in the [tex]\( y \)[/tex]-direction [tex]\( F_{\text {net }, V} \)[/tex] will be the sum of these forces, taking into account their directions:
[tex]\[
F_{\text {net }, V} = F_N - F_g - F_p \sin(32)
\][/tex]
This takes into account that [tex]\( F_N \)[/tex] is upward (positive), and both [tex]\( F_g \)[/tex] and [tex]\( F_p \sin(32) \)[/tex] are downward (negative).
Given this, you should use the equation:
[tex]\[
F_{\text {net }, V}= F_N - F_g - F_p \sin (32)
\][/tex]
Thus, the correct equation to find the net force in the [tex]\( y \)[/tex]-direction is:
[tex]\[
F_{\text {net }, V} = F_N - F_g - F_p \sin (32)
\][/tex]
