A highway curves to the left with radius of curvature of 50 m and is banked at 25° so that cars can take this curve at higher speeds.
Consider a car of mass 1692 kg whose tires have a static friction coefficient 0.89 against the pavement. How fast can the car take this curve without skidding to the outside of the curve? The acceleration of gravity is 9.8 m/3
Answer in units of m/s.
To solve this problem, we need to find the maximum speed at which the car can take the curve without skidding to the outside. This can be done by comparing the centrifugal force (outwards force experienced by the car) with the maximum static friction force (inwards force provided by the tires).
The maximum static friction force can be calculated using the equation:
Maximum static friction force = coefficient of static friction * normal force
First, we need to find the normal force exerted on the car. The normal force acts perpendicular to the surface and is equal to the weight of the car in this case since there is no vertical acceleration.
Normal force = mass * gravity
Normal force = 1692 kg * 9.8 m/s^2 = 16581.6 N
The maximum static friction force is then:
Maximum static friction force = 0.89 * 16581.6 N = 14733.024 N
Now, let's find the centrifugal force experienced by the car. The centrifugal force is the mass of the car multiplied by the centripetal acceleration, which is given by v^2 / r, where v is the velocity of the car and r is the radius of curvature.
Centrifugal force = mass * (v^2 / r)
Since the car is not skidding (assuming no horizontal external forces other than static friction), the centrifugal force is equal to the maximum static friction force.
mass * (v^2 / r) = 14733.024 N
Rearranging the equation, we get:
v^2 = maximum static friction force * r / mass
v^2 = (14733.024 N * 50 m) / 1692 kg
v^2 = 434125.28 m^2/s^2
Finally, taking the square root of both sides, we can find the maximum velocity at which the car can take the curve without skidding to the outside:
v = √(434125.28 m^2/s^2) ≈ 658.94 m/s
Therefore, the car can take the curve without skidding to the outside at a maximum speed of approximately 658.94 m/s.