To determine the total normal force acting on the bag of mulch, we need to follow these steps:
1. Understand the problem:
- A man drags a bag of mulch with a mass of 12.0 kg.
- The force applied is 39.5 N at an angle of 41.0 degrees above the horizontal.
- We aim to calculate the normal force, which is the perpendicular force exerted by the ground on the bag.
2. Determine relevant quantities:
- Mass of the bag, [tex]\( m \)[/tex] = 12.0 kg
- Applied force, [tex]\( F \)[/tex] = 39.5 N
- Angle of applied force, [tex]\( \theta \)[/tex] = 41.0^\circ
- Acceleration due to gravity, [tex]\( g \)[/tex] = 9.81 m/s²
3. Calculate the vertical component of the applied force:
The vertical component of the applied force can be determined using trigonometry. Specifically, using the sine function which gives us the vertical component when an angle is involved:
[tex]\[
F_{\text{vertical}} = F \cdot \sin(\theta)
\][/tex]
Plugging in the given values:
[tex]\[
F_{\text{vertical}} = 39.5 \, \text{N} \cdot \sin(41.0^\circ) \approx 25.91 \, \text{N}
\][/tex]
4. Calculate the weight of the bag:
The weight is the force due to gravity and is given by:
[tex]\[
\text{Weight} = m \cdot g
\][/tex]
Substituting the values:
[tex]\[
\text{Weight} = 12.0 \, \text{kg} \cdot 9.81 \, \text{m/s}² = 117.72 \, \text{N}
\][/tex]
5. Determine the normal force:
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this scenario, it is affected by the vertical component of the applied force. The normal force can be calculated by subtracting the vertical component of the applied force from the weight:
[tex]\[
n = \text{Weight} - F_{\text{vertical}}
\][/tex]
Substituting in the calculated values:
[tex]\[
n = 117.72 \, \text{N} - 25.91 \, \text{N} = 91.81 \, \text{N}
\][/tex]
Therefore, the total normal force acting upon the bag is approximately [tex]\( 91.81 \, \text{N} \)[/tex].
