Question

Banked curves are designed so that the radial component of the normal force on the car rounding the curve provides the centripetal force required to execute uniform clrcular motion and safely negotlate the curve. A car rounds a banked curve with banking angle θ-27.1° and radius of curvature 157 m.

 (a) It the coefficient of static friction between the car's tires and the road is -0.316, what is the range ot speeds for which the car can safely negotiate the turn without slipping?

 (b) What is the minimum value of for which the car's minimum safe speed is zero? Note that friction points up the incline here. 

Answer 1

For maximum speed, vehicle tends to slip upward and fiction acts in the down ward direction. Component of Normal and friction together provide the centripetal force.
Hence for radial direction, we get
N sin27.1 + mu N cos27.1 = m v²/r    (27.1 is angle of banking)
N sin27.1 + 0.316 N cos27.1 = m v²/157   .....1
In vertical direction, acceleration of vehicle is zero so
N cos27.1 - 0.316 N sin27.1 = mg   .....2
dividing equation 1 with 2 we get maximum speed
V_max² = (sin27.1 + 0.316*cos27.1)g 157 / (cos27.1 - 0.316*sin27.1)
V_max = 39 m/s

For minimum speed , vehicle tends to slip downward and friction acts in the upward direction. Sign of force of friction in equation 1 nad 2 changes. As a result ,
V_min² = (sin27.1 - 0.316*cos27.1)g 157 / (cos27.1 + 0.316*sin27.1)
v_min = 16.1 m/s

Minimum safe speed to be zero , means car should not slip down when at rest on incline.
For this mus >/= tan theta = tan 27.1 = 0.512