A 1000 kg car rounds a curve of radius 70 m banked at an angle of 12°. What is the magnitude of the friction force required for the car to travel at 82 km/h?
To find the magnitude of the friction force required for the car to travel at 82 km/h, we can analyze the forces acting on the car as it rounds the curve.
1. The gravitational force (weight) of the car acts vertically downward, with a magnitude of mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. The normal force acts perpendicular to the surface of the road. In this case, the road is banked at an angle of 12°, so the normal force can be resolved into two components:
a) The vertical component is equal to mg * cos(12°).
b) The horizontal component is equal to mg * sin(12°).
3. The friction force acts horizontally and opposes the motion of the car. The magnitude of the friction force can be determined using the equation:
Friction force = (mass of the car) * (centripetal acceleration)
Centripetal acceleration is given by the formula:
Centripetal acceleration = (velocity^2) / radius
Now we can calculate the magnitude of the friction force:
Step 1: Convert the car's velocity from km/h to m/s.
Velocity = (82 km/h) * (1000 m/km) * (1 h/3600 s)
Velocity = 22.78 m/s
Step 2: Calculate the centripetal acceleration.
Centripetal acceleration = (22.78 m/s)^2 / 70 m
Centripetal acceleration = 7.48 m/s^2
Step 3: Determine the magnitude of the friction force.
Friction force = (1000 kg) * (7.48 m/s^2)
Friction force = 7480 N
Therefore, the magnitude of the friction force required for the car to travel at 82 km/h is 7480 Newtons.