Final answer:
The minimum coefficient of static friction for a car navigating a banked curve is found by using the car's mass, velocity, curve radius, and banking angle to calculate the balance of forces that prevent skidding. The equation considers gravitational force, normal force, and frictional force.
Step-by-step explanation:
To find the minimum coefficient of static friction (μs) for a 1050 kg car navigating a curve with a radius of 95 m, banked at an angle of θ=10.8°, at a speed of 70 km/h without skidding, we need to consider the forces acting on the car. These forces are the gravitational force, the normal force from the road, and the frictional force that opposes sliding.
First, convert the speed to meters per second by multiplying by (1000 m/km)/(3600 s/h), which gives approximately 19.44 m/s. The gravitational force is equal to mg, where m is the car's mass and g is the acceleration due to gravity (9.8 m/s²). The normal force can be broken into two components: one supporting the weight of the car and one providing the centripetal force to keep the car moving in a circle. The friction force must provide the difference between the component of gravitational force down the bank and the centripetal force needed.
The equation that relates these forces is:
μs = ∠F / N
∠F is the net force towards the center, and N is the normal force perpendicular to the surface. For a banked curve without slipping:
∠F = mv² / r
where m is the mass, v is the velocity, and r is the radius of the curve. The normal force N is mg/cos(θ). To find μs, we can rearrange the formula to:
μs ≥ (mv²/r - mg tan(θ)) / (mg cos(θ))
After calculating, we can find the minimum coefficient of static friction needed to prevent the car from skidding on the curve.