A race car travels 40 m/s around a banked (45° with the horizontal) circular (radius = 0.20 km)
track. What is the magnitude of the resultant force on the 80-kg driver of this car?
640
To find the magnitude of the resultant force on the driver of the car, we need to consider the forces acting on them. In this case, the main forces are the gravitational force and the centripetal force.
1. Gravitational force:
The gravitational force acting on the driver can be calculated using the equation: Fg = mg, where m is the mass of the driver and g is the acceleration due to gravity. Given that the mass of the driver (m) is 80 kg and the acceleration due to gravity (g) is 9.8 m/s^2, we can calculate the gravitational force.
Fg = (80 kg) * (9.8 m/s^2)
Fg = 784 N
2. Centripetal force:
The centripetal force is the force that keeps the car moving in a circular path. It is given by the equation: Fc = (m * v^2) / r, where m is the mass of the car, v is the velocity, and r is the radius of the circular path. Given that the velocity (v) is 40 m/s and the radius (r) is 0.20 km (which needs to be converted to meters), we can calculate the centripetal force.
Fc = (80 kg) * (40 m/s)^2 / (0.20 km * 1000 m/km)
Fc = 64000 N
3. Resultant force:
The resultant force on the driver is the vector sum of the gravitational force and the centripetal force. The magnitude of the resultant force can be calculated using the Pythagorean theorem since the two forces act at right angles to each other (assuming the banking angle is sufficient to provide all the necessary centripetal force).
Fresultant = √(Fg^2 + Fc^2)
Fresultant = √(784 N)^2 + (64000 N)^2
Fresultant ≈ 64234.08 N
Therefore, the magnitude of the resultant force on the 80-kg driver of this car is approximately 64234.08 N.
To find the magnitude of the resultant force on the driver of the race car, we first need to consider the forces acting on the driver while driving around the banked circular track.
The forces acting on the driver can be broken down into two components: the gravitational force (mg) and the centripetal force (Fc). The gravitational force acts vertically downward and is given by the formula Fg = mg, where m is the mass of the driver (80 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). The centripetal force acts inward and is responsible for keeping the car moving in a circular path.
The centripetal force on a banked circular track is provided by the friction between the tires of the car and the track surface. It can be calculated using the formula Fc = mv^2/r, where m is the mass of the car and driver (80 kg), v is the velocity of the car (40 m/s), and r is the radius of the circular track (0.20 km = 200 m).
Now let's calculate the magnitude of the resultant force on the driver.
1. Calculate the centripetal force:
Fc = mv^2/r
= (80 kg)(40 m/s)^2 / 200 m
= 6400 kg·m/s^2 / 200 m
= 32,000 N
2. Calculate the gravitational force:
Fg = mg
= (80 kg)(9.8 m/s^2)
= 784 N
3. Determine the magnitude of the resultant force:
The resultant force is the vector sum of the gravitational force and the centripetal force. Since the two forces act in different directions, we need to find the net force.
Since the track is banked at a 45° angle, the vertical component of the centripetal force (Fcv) cancels out the gravitational force (Fg). Therefore, the net force acting on the driver is the horizontal component of the centripetal force (Fch), which is given by Fch = Fc * cos(45°).
Fch = 32,000 N * cos(45°)
= 32,000 N * 0.7071
≈ 22,628.68 N
So, the magnitude of the resultant force on the 80-kg driver of this car is approximately 22,629 N.