You are driving on a circular ramp of radius 69.0m on an icy January day. The ice-covered ramp is banked at a 18.0° angle, and you anticipate that the ramp is not going to provide any significant friction to help keep your car on the road.
(a) What must your speed be so that your car does not slide off the road?
(b) What are the direction and magnitude of the net force acting on you at that point? Your mass is 80.0kg .
To determine the speed needed to prevent your car from sliding off the road on the icy ramp, we can analyze the forces acting on your car.
Let's start by considering the forces in the vertical direction (perpendicular to the ramp). There are two forces acting: the gravitational force (mg) and the normal force (N). The component of the normal force in the vertical direction cancels out the gravitational force, keeping the car from sinking into the ramp.
The normal force (N) can be determined using the formula:
N = mg * cosθ
Where:
m = mass of the car = 80.0 kg
g = acceleration due to gravity = 9.8 m/s^2
θ = angle of inclination = 18.0° or 18.0° * (π/180) rad (converting to radians)
N = 80.0 kg * 9.8 m/s^2 * cos(18.0°) ≈ 741.43 N
Now, let's analyze the forces in the horizontal direction (parallel to the ramp). In this case, there are two forces acting: the friction force (f) and the horizontal component of the normal force (N_x). The friction force is responsible for keeping the car from sliding off the road.
The friction force (f) can be determined using the formula:
f = μ * N
Where:
μ = coefficient of friction (equal to zero in this case, as there is no significant friction on the icy ramp)
N = component of the normal force in the horizontal direction = N * sinθ
f = 0 (since μ = 0)
Since the friction force is zero, there is no force acting to prevent the car from sliding off the road. Therefore, your speed is not relevant in this case. The answer to part (a) is there is no minimum speed required to prevent sliding off the road.
Moving on to part (b):
Since the net force is the vector sum of all forces acting on the car, and the only force acting on the car is in the vertical direction (gravitational force), the net force is in the vertical direction. The magnitude of the net force (F_net) can be determined using the formula:
F_net = N_y = N * sinθ
N * sinθ = 741.43 N * sin(18.0°) ≈ 234.53 N
The direction of the net force is downward, opposite to the upward normal force, indicating that it is responsible for providing the centripetal force needed to keep the car moving in a circular path.
Therefore, the answer to part (b) is:
The net force acting on you at that point is approximately 234.53 N, directed downward (perpendicular to the ramp).