Physics Mechanics?

Question

Answers in brackets, How do you get to the answers? tnx

A car goes around a curved stretch of flat roadway of radius R = 114.0 m. The magnitudes of the horizontal and vertical components of force the car exerts on a securely seated passenger are, respectively, X = 205.0 N and Y = 552.0 N.

q.What is the minimum coefficient of static friction between the tyres and the road needed to negotiate this turn without sliding out? (0.371)

This stretch of highway is a notorious hazard during the winter months when it can be quite slippery. Accordingly the LTSA decides to bank it at an angle φ = 24.0 ° to the horizontal.

q. At what speed could the car now negotiate this curve without needing to rely on any frictional force to prevent it slipping upwards or downwards on the banked surface?(80.3)

q. Suppose the above car negotiates the banked curve at 100 km/h maximum safe speed. What are the horizontal and vertical force components this car now exerts on the seated passenger? (383 and 582)

Answer 1

The angle A = arcTan(Fx/Fy)

Force of friction f=uN
and also f=T
T- force parallel to the surface
N - force normal to the surface
N= mg cos (A)
T= mg sin(A)
m - mass of car and passengers
g - acceleration due to gravity
we have
mg sin(A) = u mg cos (A) then

u = mg sin(A) /mg cos (A)= tan(A)
and from above tan (A)= Fx/Fy

u=Fx/Fy=552/205=0.371

(b) The centripetal force Fc=ma
a= v^2/R
Fc=mv^2 / R ( normal to gravitational force!)
Let's write the force equation (p=24 degrees)
Now
N=W cos (P) + Fc sin(P) and
f- force of friction f=0 since it is slippery
Fd - dowward component of car + pasengers weight
Since f=uN the component responsible for forsing the car off the road is Fc cos(P)
Fc cos(P)=Wsin(P) + u N
(mv^2/R ) cos(P) = mg sin(P) + u mg cos (P) + u(mv^2/R ) sin(P))
(v^2/R ) cos(P) = g sin(P) + u g cos (P) + u(v^2/R ) sin(P))
(v^2/R ) cos(P) - u(v^2/R ) sin(P)) = g (sin(P) + u cos(P))
(v^2 ) [cos(P) - u sin(P)] = g R(sin(P) + u cos(P))
(v^2 ) = [g R(sin(P) + u cos(P))] / [cos(P) - u sin(P)]
v=sqrt([9.81 x 114.0 (sin(24) + u cos(24))] / [cos(24) - u sin(24)])

if u= 0

v=sqrt([9.81 x 114.0 (sin(24) )] / [cos(24) ]) = 22.3 m/s
or v = 80.2 km/hour

If you understood part (b) the part (c) is easy.