To determine the net force in the [tex]\( y \)[/tex]-direction when a box is pushed down at an angle of [tex]\( 32^\circ \)[/tex] on a rough surface, we need to consider the forces acting in the vertical direction.
Here are the forces to consider:
1. Normal Force ([tex]\( F_N \)[/tex]): This is the force exerted by the surface perpendicular to the box.
2. Gravitational Force ([tex]\( F_g \)[/tex]): This is the weight of the box acting downward.
3. Force due to the push ([tex]\( F_p \)[/tex]): This force is being applied at an angle. We need to decompose this force into its vertical and horizontal components. Since the push is angled downward, the vertical component of this force can be found using [tex]\( F_p \sin(32^\circ) \)[/tex].
Given these considerations, we can write down the expression for the net force in the vertical direction:
[tex]\[ F_{\text{net}, y} = F_N - F_g - F_p \sin(32^\circ) \][/tex]
Here is a step-by-step breakdown:
1. Normal Force ([tex]\( F_N \)[/tex]): Acts upwards, perpendicular to the rough surface.
2. Gravitational Force ([tex]\( F_g \)[/tex]): Acts downwards, equal to the weight of the box.
3. Vertical Component of the Push Force [tex]\( (F_p \sin(32^\circ)) \)[/tex]: Acts downward, since the force is applied at an angle downward with respect to the horizontal.
Combining these forces, the upward force is [tex]\( F_N \)[/tex] and the downward forces are [tex]\( F_g \)[/tex] and [tex]\( F_p \sin(32^\circ) \)[/tex].
Thus, the correct equation to find the net force in the [tex]\( y \)[/tex]-direction is:
[tex]\[ F_{\text{net}, y} = F_N - F_g - F_p \sin(32^\circ) \][/tex]
Therefore, the correct answer is:
[tex]\[ F_{\text{net}, y} = F_N - F_g - F_p \sin(32^\circ) \][/tex]
