A car of weight W = 12.3 kN makes a turn on a track that is banked at an angle of θ = 20.9°. Inside the car, hanging from a short string tied to the rear-view mirror, is an ornament. As the car turns, the ornament swings out at an angle of φ = 28.3° measured from the vertical inside the car. What is the force of static friction between the car and the road?
To find the force of static friction between the car and the road, we need to consider the forces acting on the car as it makes a turn on the banked track. One of these forces is the static friction force.
Let's break down the forces acting on the car:
1. Weight: The weight of the car acts downwards, vertically towards the ground. The magnitude of the weight is given as W = 12.3 kN.
2. Normal Force: The normal force acts perpendicular to the inclined surface of the banked track. It can be decomposed into two components – one pointing perpendicular to the inclined surface (N⊥) and the other parallel to the inclined surface of the track (N∥). The component that comes into play in this problem is N∥.
3. Centripetal Force: The car makes a turn on the banked track, which requires a centripetal force directed towards the center of the turn. This force is provided by the static friction force.
Now, let's find the value of the static friction force.
We can start by finding the components of the normal force:
N⊥ = W * cos(θ)
N∥ = W * sin(θ)
Next, we find the horizontal component of the centripetal force (Fc∥) using Newton's second law:
Fc∥ = m * a∥
Since the car is moving in a circular path, the acceleration can be expressed as a∥ = v² / R, where v is the speed of the car and R is the radius of the turn.
Now, we know that the horizontal component of the centripetal force is provided by the static friction:
Fc∥ = Fs = µs * N∥
where µs is the coefficient of static friction.
Thus, we can set up the following equation:
µs * N∥ = m * v² / R
Substituting the expressions for N∥ and solving for µs:
µs * (W * sin(θ)) = m * v² / R
µs = (m * v²) / (W * sin(θ))
Now, we have all the necessary values to calculate the force of static friction between the car and the road.
To find the force of static friction between the car and the road, we need to consider the forces acting on the car as it makes the turn.
Let's analyze the forces acting on the car:
1. Weight (W): The weight of the car acts vertically downward with a magnitude of 12.3 kN.
2. Normal Force (N): The normal force is the force exerted by the road on the car perpendicular to the surface. It acts upward and is equal in magnitude to the vertical component of the weight of the car, which can be calculated as N = W * cos(θ).
3. Frictional Force (Ff): The frictional force acts horizontally and opposes the tendency of the car to slip. It acts toward the center of the circular path the car is following.
To maintain equilibrium and prevent slipping, the vertical component of the weight of the car must be balanced by the normal force (N), and the horizontal component of the weight (W * sin(θ)) must be balanced by the frictional force (Ff).
Now, let's calculate the values:
Vertical Component of Weight (Wv) = W * cos(θ)
Wv = 12.3 kN * cos(20.9°)
Wv ≈ 11.393 kN
Frictional Force (Ff) = W * sin(θ)
Ff = 12.3 kN * sin(20.9°)
Ff ≈ 4.288 kN
Finally, the force of static friction between the car and the road is approximately 4.288 kN.