To determine the force exerted on the roller coaster car by the track at the bottom of the dip, we need to calculate the centripetal force and the gravitational force acting on the car, and then sum these forces.
### Step-by-Step Solution:
#### 1. Given Data:
- Mass of the roller coaster car, [tex]\( m = 500 \)[/tex] kg
- Speed of the car at the dip, [tex]\( v = 18 \)[/tex] meters/second
- Radius of curvature of the track, [tex]\( r = 12 \)[/tex] meters
- Acceleration due to gravity, [tex]\( g = 9.8 \)[/tex] meters/second[tex]\(^2\)[/tex]
#### 2. Calculate the Centripetal Force:
The centripetal force is necessary to keep the roller coaster car moving in a circular path and is directed towards the center of curvature. It can be calculated using the formula:
[tex]\[ F_c = \frac{m \cdot v^2}{r} \][/tex]
Substitute the given values into the formula:
[tex]\[ F_c = \frac{500 \cdot 18^2}{12} \][/tex]
Perform the calculations inside the formula:
[tex]\[ F_c = \frac{500 \cdot 324}{12} \][/tex]
[tex]\[ F_c = \frac{162000}{12} \][/tex]
[tex]\[ F_c = 13500 \text{ N} \][/tex]
The centripetal force acting on the roller coaster car is 13,500 Newtons (N).
#### 3. Calculate the Gravitational Force:
The gravitational force acting on the roller coaster car is given by the weight of the car. This can be calculated using the formula:
[tex]\[ F_g = m \cdot g \][/tex]
Substitute the values:
[tex]\[ F_g = 500 \cdot 9.8 \][/tex]
Perform the calculation:
[tex]\[ F_g = 4900 \text{ N} \][/tex]
The gravitational force acting on the roller coaster car is 4,900 Newtons (N).
#### 4. Determine the Total Force Exerted by the Track:
At the bottom of the dip, the force exerted by the track must counteract both the centripetal force and the gravitational force. Since these forces have the same direction (both acting upwards towards the center of the circular path and counteracting weight), the total force exerted by the track is the sum of the centripetal force and the gravitational force:
[tex]\[ F_{\text{total}} = F_c + F_g \][/tex]
Substitute the calculated values:
[tex]\[ F_{\text{total}} = 13500 + 4900 \][/tex]
[tex]\[ F_{\text{total}} = 18400 \text{ N} \][/tex]
### Conclusion:
The total force exerted by the track on the roller coaster car at the bottom of the dip is 18,400 Newtons (N).
