To find the normal force acting on the barrel, we need to consider the various forces involved. These forces include the gravitational force (weight) acting downward, the vertical component of the applied force acting upward, and the resulting normal force.
Here is a step-by-step breakdown:
1. Calculate the weight of the barrel:
The weight ([tex]\( W \)[/tex]) of the barrel can be calculated using the formula:
[tex]\[
W = m \cdot g
\][/tex]
where [tex]\( m \)[/tex] is the mass of the barrel and [tex]\( g \)[/tex] is the acceleration due to gravity ([tex]\( 9.81 \, m/s^2 \)[/tex]).
Substituting the given values:
[tex]\[
W = 14.5 \, \text{kg} \times 9.81 \, m/s^2 = 142.245 \, \text{N}
\][/tex]
2. Calculate the vertical component of the applied force:
The vertical component of the applied force ([tex]\( F_{\text{vertical}} \)[/tex]) can be found using the sine of the given angle ([tex]\( \theta \)[/tex]):
[tex]\[
F_{\text{vertical}} = F \cdot \sin(\theta)
\][/tex]
where [tex]\( F \)[/tex] is the magnitude of the applied force and [tex]\( \theta \)[/tex] is the angle in degrees.
First, convert the angle from degrees to radians:
[tex]\[
\theta_{\text{radians}} = \frac{48.7^\circ \times \pi}{180} = 0.849975 \, \text{radians}
\][/tex]
Now, calculate the vertical component:
[tex]\[
F_{\text{vertical}} = 22.1 \, \text{N} \cdot \sin(0.849975) = 16.6029 \, \text{N}
\][/tex]
3. Calculate the normal force:
The normal force ([tex]\( N \)[/tex]) is the force exerted by the ground on the barrel that counteracts the weight and the vertical component of the applied force:
[tex]\[
N = W - F_{\text{vertical}}
\][/tex]
Substituting the calculated values:
[tex]\[
N = 142.245 \, \text{N} - 16.6029 \, \text{N} = 125.642 \, \text{N}
\][/tex]
Therefore, the normal force acting upon the barrel is [tex]\(\boxed{125.642 \, \text{N}}\)[/tex].
