Question

A car going at v = 29.7 m/s (67 mph) rounds a curve of radius R = 50.0 m, where the road is banked at an angle of θ = 30.0°. What is the minimum that the coefficient of static friction between the tires and the asphalt must be so that the car can go through the curve without skidding?

Answer 1

μ = 0.6

Step-by-step explanation:

given,

speed of car = 29.7 m/s

Radius of curve = 50 m

θ = 30.0°

minimum static friction = ?

now,

writing all the forces acting along y-direction

N cos θ - f sinθ = mg

N cos θ -μN sinθ = mg

`N = (m g)/(cos\theta-\mu sin \theta)`

now, writing the forces acting along x- direction

N sin θ + f cos θ = F_{net}

N cos θ + μN sinθ = F_{net}

`(m g)/(cos\theta-\mu sin \theta)(cos \theta + \mu sin\theta)=F_(net)`

taking cos θ from nominator and denominator

`F_(net) =(tan\theta + \mu)/(1-\mutan\theta). mg`

`(mv^2)/(r)=(tan\theta + \mu)/(1-\mutan\theta). mg`

`(v^2)/(r)=(tan\theta + \mu)/(1-\mutan\theta)g`

`\mu=(v^2 -r g tan\theta)/(v^2tan\theta + r g)`

now, inserting all the given values

`\mu=(29.7^2 -50 * 9.8tan 30^0)/(29.7^2* tan 30^0 +50 * 9.8)`

μ = 0.6