Question

A curve of radius 69 mm is banked for a design speed of 76 km/hkm/h .

Part A If the coefficient of static friction is 0.28 (wet pavement), at what range of speeds can a car safely make the curve? [Hint: Consider the direction of the friction force when the car goes too slow or too fast.] Express your answers using two significant figures. Enter your answers numerically separated by a comma.

Answer 1

v = 76 km/h = 21.111 m/s

so,

the banking angle is found as

= arctan ( v² / gr)

= arctan ( 21.111² / 9.8 * 69)

= 33.388 degree

Now

there are two cases

Case 1 - when car slides upwards

so,

Ncos = mg + fsin

Ncos = mg + uNsin

N = mg / ( cos - u sin )

Fnet = Nsin + uNcos

put the value of N, we get

Fnet = tan + u mg / ( 1 - utan)

set this equal to centripetal force acting

so,

tan + u mg / ( 1 - utan) = mv² / r

so,

v = sqrt ( tan + u / (1 - utan)) * (rg) )

put in the values

v = sqrt ( tan 33.388 + 0.28 / (1 - 0.28 * tan 33.388)) * (69 * 9.8)

v = 27.9 m/s

Case 2 - when slides downwards

Now, friction force acts up the incline

using same approach, I got

Ncos + fsin = mg

N cos + uNsin = mg

N = mg / ( cos + u sin )

so,

Fnet = tan - u mg / ( 1 + utan)

so,

v = sqrt ( tan - u / (1 + utan)) * (rg) )

v = sqrt ( tan 33.388 - 0.28 / (1 + 0.28 * tan 33.388)) * (69 * 9.8))

v = 14.71 m/s