Question

A curve of radius 75 m is banked for a design speed of 100 km/h .

If the coefficient of static friction is 0.36 (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.]

Answer 1

given

r = 75 m

V = 100 km/h

= 27.77 m/sec

= the coefficient of static friction is 0.36

using equation

tan = V² / r g

= 27.77² / 75 x 9.8

= 46.375^o

the normal acceleration is a_n = g cos + ( v² sin / r )

and the friction acceleration is a_f = a_n

then the maximum speed at gravity

( g sin ) + ( g cos + ( v² sin / r ) ) = v² cos / r

( 9.8 x sin46.375 ) + 0.36 x ( 9.8 x cos46.375 + ( v² sin46.375 / 75 ) ) = v² cos46.375 / 75

( 7.093 ) + 0.36 x ( 6.761 + ( v² x 0.00965 ) ) = v² x 0.00919

7.093 + 2.4339 + 0.003474 v² = v² x 0.00919

v² = 9.5269 / 0.005716

v = 40.8253 m/sec

or

v =146.97 km/h

and the minimum speed at gravity

( g sin ) - ( g cos + ( v² sin / r ) ) = v² cos / r

( 9.8 x sin46.375 ) - 0.36 x ( 9.8 x cos46.375 + ( v² sin46.375 / 75 ) ) = v² cos46.375 / 75

( 7.093 ) - 0.36 x ( 6.761 + ( v² x 0.00965 ) ) = v² x 0.00919

7.093 - 2.4339 - 0.003474 v² = v² x 0.00919

v² = 4.6591 / 0.012664

v = 19.1807 m/sec

or

v =69.05 km/h

so the range of speeds can a car safely make the curve is 146.97 km/h to 69.05 km/h