Let's walk through the problem step-by-step to find the required forces and the acceleration of the sled.
### 1. Calculate the weight of the sled
The weight of the sled can be determined using the formula:
[tex]\[ \text{Weight} = \text{mass} \times \text{gravity} \][/tex]
Given:
- Mass of the sled, [tex]\( \text{mass\_sled} = 30.0 \, \text{kg} \)[/tex]
- Acceleration due to gravity, [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
Therefore:
[tex]\[ \text{Weight} = 30.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 294.0 \, \text{N} \][/tex]
So, the weight of the sled is:
[tex]\[ \boxed{294.0 \, \text{N}} \][/tex]
### 2. Calculate the normal force exerted on the sled
To determine the normal force, we need to account for the vertical component of the pulling force, which affects the normal force since it lifts the sled partially.
First, calculate the vertical component of the pulling force:
[tex]\[ \text{Pulling Force (Vertical)} = 12.0 \, \text{N} \times \sin(45^\circ) \][/tex]
Since [tex]\( \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707 \)[/tex]:
[tex]\[ \text{Pulling Force (Vertical)} = 12.0 \, \text{N} \times 0.707 = 8.484 \, \text{N} \][/tex]
The normal force is the weight of the sled minus the vertical component of the pulling force:
[tex]\[ \text{Normal Force} = \text{Weight} - \text{Pulling Force (Vertical)} \][/tex]
Therefore:
[tex]\[ \text{Normal Force} = 294.0 \, \text{N} - 8.484 \, \text{N} = 285.516 \, \text{N} \][/tex]
Rounding to the nearest whole number:
[tex]\[ \boxed{286 \, \text{N}} \][/tex]
### 3. Calculate the acceleration of the sled
To determine the acceleration, we need the net horizontal force and then use Newton's second law.
First, calculate the horizontal component of the pulling force:
[tex]\[ \text{Pulling Force (Horizontal)} = 12.0 \, \text{N} \times \cos(45^\circ) \][/tex]
Since [tex]\( \cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707 \)[/tex]:
[tex]\[ \text{Pulling Force (Horizontal)} = 12.0 \, \text{N} \times 0.707 = 8.484 \, \text{N} \][/tex]
Next, compute the net horizontal force by summing the horizontal components of all forces and subtracting the friction force:
[tex]\[ \text{Net Horizontal Force} = \text{Pulling Force (Horizontal)} + \text{Pushing Force} - \text{Friction Force} \][/tex]
Given:
- Pushing Force = [tex]\( 8.00 \, \text{N} \)[/tex]
- Friction Force = [tex]\( 5.00 \, \text{N} \)[/tex]
[tex]\[ \text{Net Horizontal Force} = 8.484 \, \text{N} + 8.00 \, \text{N} - 5.00 \, \text{N} = 11.484 \, \text{N} \][/tex]
Finally, use Newton's second law to find the acceleration:
[tex]\[ \text{Acceleration} = \frac{\text{Net Horizontal Force}}{\text{Mass}} \][/tex]
Therefore:
[tex]\[ \text{Acceleration} = \frac{11.484 \, \text{N}}{30.0 \, \text{kg}} = 0.3828 \, \text{m/s}^2 \][/tex]
Rounding to the nearest hundredth:
[tex]\[ \boxed{0.38 \, \text{m/s}^2} \][/tex]
