If a curve with a radius of 80m is properly banked for a car traveling 63km/h , what must be the coefficient of static friction for a car not to skid when traveling at 97km/h ?
To determine the coefficient of static friction for a car not to skid when traveling at a certain speed, we can use the concept of banking of curves. When a curve is properly banked, it helps to provide the necessary centripetal force for the vehicle to navigate the curve without relying solely on friction.
Let's start by finding the angle of banking (θ) that corresponds to the given conditions. The formula to determine the banking angle is:
θ = arctan(v^2 / (g * r))
Where:
- θ is the angle of banking,
- v is the speed of the car,
- g is the acceleration due to gravity (approximated as 9.8 m/s^2),
- r is the radius of the curve.
First, let's convert the given speeds into meters per second (m/s):
- For the car traveling at 63 km/h:
v1 = 63 km/h * (1000 m / 1 km) * (1 h / 3600 s) = 17.5 m/s
- For the car traveling at 97 km/h:
v2 = 97 km/h * (1000 m / 1 km) * (1 h / 3600 s) = 26.9 m/s
Now, let's calculate the angle of banking for the two speeds:
θ1 = arctan((17.5 m/s)^2 / (9.8 m/s^2 * 80 m))
θ1 ≈ arctan(3.98) ≈ 75.7 degrees
θ2 = arctan((26.9 m/s)^2 / (9.8 m/s^2 * 80 m))
θ2 ≈ arctan(9.29) ≈ 83.5 degrees
To find the coefficient of static friction (µ) required for the car not to skid at 97 km/h, we need to determine when the component of the gravitational force along the surface of the road (mg * sin θ) equals the centripetal force required to keep the car on the road.
The equation for the centripetal force is:
F_c = m * v^2 / r
The maximum static friction force (F_f) is given by:
F_f = µ * m * g
When the car is not skidding, F_f can be equal to F_c.
Therefore, µ * m * g = m * v^2 / r
Cancelling out the mass (m) from both sides of the equation, we get:
µ * g = v^2 / r
Now, let's calculate the required coefficient of static friction (µ2) for the car traveling at 97 km/h:
µ2 = (v2^2 / r) / g
µ2 = (26.9 m/s)^2 / (80 m) / (9.8 m/s^2)
µ2 ≈ 0.91
Therefore, the coefficient of static friction must be approximately 0.91 for the car not to skid when traveling at 97 km/h.
Keep in mind that this calculation assumes other factors, such as road conditions, tire grip, and vehicle dynamics, remain constant.