A car drives around a curve with radius 410 m at a speed of 39 m/s. The road is banked at 4.6°. The mass of the car is 1600 kg.
(a) What is the frictional force on the car?
(b) At what speed could you drive around this curve so that the force of friction is zero?
To determine the frictional force on the car, we need to consider the forces acting on the car as it moves around the curved road. There are two main forces at play: the force of gravity pulling the car downward, and the normal force acting perpendicular to the road surface.
(a) To calculate the frictional force on the car, we first find the net force acting towards the center of the curve. This force is provided by the component of the gravitational force perpendicular to the road surface. The force of gravity can be calculated using the equation: F_gravity = mass * acceleration due to gravity.
F_gravity = 1600 kg * 9.8 m/s^2 = 15,680 N
Next, we need to find the net gravitational force acting towards the center of the curve. This force can be found by multiplying the gravitational force by the sine of the angle of the banked road: F_net_gravity = F_gravity * sin(banked angle).
F_net_gravity = 15,680 N * sin(4.6°) ≈ 1,158 N
Since the car is moving in a curved path, the net force towards the center of the curve is provided by the frictional force. Therefore, the frictional force can be calculated by multiplying the mass of the car by the centripetal acceleration: F_friction = mass * centripetal acceleration.
The centripetal acceleration can be found using the formula: centripetal acceleration = (velocity^2) / radius.
Centripetal acceleration = (39 m/s)^2 / 410 m ≈ 3.719 m/s^2
F_friction = 1600 kg * 3.719 m/s^2 ≈ 5,950 N
Therefore, the frictional force on the car is approximately 5,950 Newtons.
(b) To determine the speed at which the force of friction is zero, we need to consider the point where the net gravitational force is equal to the needed centripetal force for circular motion. In this case, the centripetal force is zero, meaning the frictional force is zero.
F_net_gravity = 15,680 N * sin(4.6°) ≈ 1,158 N
Setting this equal to zero, we can solve for the velocity:
(39 m/s)^2 / 410 m = 9.8 m/s^2 * sin(4.6°)
By solving this equation, we can find the velocity at which the frictional force is zero.