To determine the total normal force acting on the bag, let's break the problem into manageable steps and analyze the forces involved.
### 1. Calculate the Vertical Component of the Applied Force
The applied force of 39.5 N is at an angle of [tex]\( 41.0^{\circ} \)[/tex]. The vertical component of this force can be found using trigonometry:
[tex]\[
F_{\text{vertical}} = F \cdot \sin(\theta)
\][/tex]
where [tex]\( F \)[/tex] is the applied force and [tex]\( \theta \)[/tex] is the angle.
Using:
[tex]\[ F = 39.5 \, \text{N} \][/tex]
[tex]\[ \theta = 41.0^{\circ} \][/tex]
[tex]\[ F_{\text{vertical}} = 39.5 \cdot \sin(41.0^{\circ}) \][/tex]
After calculating, we get:
[tex]\[ F_{\text{vertical}} \approx 25.914 \, \text{N} \][/tex]
### 2. Calculate the Weight of the Bag
The weight of the bag is the force due to gravity and can be calculated using:
[tex]\[
W = m \cdot g
\][/tex]
where [tex]\( m \)[/tex] is the mass and [tex]\( g \)[/tex] is the acceleration due to gravity.
Given:
[tex]\[ m = 12.0 \, \text{kg} \][/tex]
[tex]\[ g = 9.8 \, \text{m/s}^2 \][/tex]
[tex]\[ W = 12.0 \cdot 9.8 \][/tex]
[tex]\[ W \approx 117.6 \, \text{N} \][/tex]
### 3. Determine the Normal Force
The normal force is the force exerted by the surface to support the weight of the bag and balance any other vertical forces. Since the bag is moving at a constant speed, the vertical forces must balance out. The weight of the bag [tex]\( W \)[/tex] is opposed by the normal force [tex]\( n \)[/tex] and the vertical component of the applied force [tex]\( F_{\text{vertical}} \)[/tex].
The normal force [tex]\( n \)[/tex] can be calculated as:
[tex]\[
n = W - F_{\text{vertical}}
\][/tex]
Substituting the known values:
[tex]\[ n = 117.6 - 25.914 \][/tex]
[tex]\[ n \approx 91.686 \, \text{N} \][/tex]
Therefore, the total normal force acting upon the bag is approximately:
[tex]\[
\boxed{91.686 \, \text{N}}
\][/tex]
