Question

A highway curve with a radius of 750 m is banked properly for a car traveling 120 km/h. If a 1600- kg Porshe 928S rounds the curve at 230 km/h, how much sideways force must the tires exert against the road if the car does not skid?

Answer 1

Here
mass of porshe = M = 1600 kg
radius of path = 750m
Vp = 230 km/hr = 63.88 m/s
Vmax for Banked road = 120 km/hr = 120/3.6 = 33.33 m/s

Writing Different force actign on car while moving on banking road,
From Newton law of motion
ΣF(x) = 0

F(x) = mv²/r = nsinθ -------------------->(1)
F(y) = ncosθ - mg = 0 -------------------->(2)

Subtituting 2 in 1

mv^2/r = (mg/cosθ)sinθ

θ = arcTan(V²/(g*r) )
θ = arcTan(33.33^2/(9.8*750) )
θ = 8.59 degrees

Since 22.2m/s is the critical speed, anything over this speed will require friction to keep the car from sliding up .
again using Newton Second law of motion we get

F(x) = mv²/r = nsinθ + fcosθ----------------------->(3)

F(y) = 0 = nsinθ - mg - fsinθ
n = (mg + fsinθ) / cosθ------------------------------>(4)
Using 4 in 3 and writing expression for f we get,

f = m(v^2*cosθ - g*r*sinθ)/ ( r*(sin²θ + cos²θ) )
f = 1600 * (63.88^2*cos8.59 - 9.8*750*sin8.59)/ ( 750*( (sin8.59)^2 + (Cos8.59)^2 )
f = 1600 * (63.88^2*0.988 - 9.8*750*0.149)/ ( 750*( (0.149)^2 + (0.988)^2 )
f = 6274.994 N

6274.994 N sideways force must the tires exert against the road to not to skid