Answer:
1. ![3.4^{o}](https://tex.z-dn.net/?f=3.4%5E%7Bo%7D)
2. 163.3 m
Explanation:
Static friction between road and rubber, μs =0.06
The maximum speed of the car, v = 50 km/h
= (50)(1000/3600) m/s
= 13.89 m/s
The acceleration due to gravity, ![g = 9.81 m/s^{2}](https://tex.z-dn.net/?f=g%20%3D%209.81%20m%2Fs%5E%7B2%7D)
The frictional force, f = μsN ...... (1)
The component mg cosθ which balance the normal reaction N
The component mg sinθ acts in an opposite direction to the frictional force f.
ΣF = mg sinθ-f = 0 ...... (2)
Substitute the equation (1) in equation (2), we get
ΣF = mgsinθ-μsN = 0
mgsinθ-μsmgcosθ =0
μs = sinθ/cosθ
tanθ = μs
θ = tan-1( μs) = tan-1(0.06) = ![3.4^{o}](https://tex.z-dn.net/?f=3.4%5E%7Bo%7D)
(b)The vertical component of the force is
N cosθ = fsinθ+mg
N cosθ = μsNsinθ+mg
N[cosθ- μs sinθ] = mg ...... (3)
The horizontal component of the force along the motion of the car is
Nsinθ+fcosθ = ma (Centripetal acceleration, ![a = \frac {v^{2}}{r}](https://tex.z-dn.net/?f=a%20%3D%20%5Cfrac%20%7Bv%5E%7B2%7D%7D%7Br%7D)
Nsinθ+fcosθ = ![m(\frac {v^{2}}{r})](https://tex.z-dn.net/?f=m%28%5Cfrac%20%7Bv%5E%7B2%7D%7D%7Br%7D%29)
Nsinθ+μsNcosθ = ![m(\frac {v^{2}}{r})](https://tex.z-dn.net/?f=m%28%5Cfrac%20%7Bv%5E%7B2%7D%7D%7Br%7D%29)
N[sinθ+μs cosθ] =
...... (4)
Dividing the equation (4) with equation (3),
[sinθ+μscosθ]/[cosθ- μs sinθ] = ![\frac {v^{2}}{rg}](https://tex.z-dn.net/?f=%5Cfrac%20%7Bv%5E%7B2%7D%7D%7Brg%7D)
cosθ[sinθ/cosθ+μs]/cosθ[1- μs sinθ/cosθ] =![\frac {v^{2}}{rg}](https://tex.z-dn.net/?f=%5Cfrac%20%7Bv%5E%7B2%7D%7D%7Brg%7D)
[tanθ+μs]/[1-μs tanθ] =
From part (1), tanθ = μs
Then the above equation becomes
![\frac {(\mu_s+\mu_s]}{[1-\mu_s^{2}]} =\frac {v^{2}}{rg}](https://tex.z-dn.net/?f=%5Cfrac%20%7B%28%5Cmu_s%2B%5Cmu_s%5D%7D%7B%5B1-%5Cmu_s%5E%7B2%7D%5D%7D%20%3D%5Cfrac%20%7Bv%5E%7B2%7D%7D%7Brg%7D)
![\frac {(2\mu_s]}{[1-\mu_s^{2}]} =\frac {v^{2}}{rg}](https://tex.z-dn.net/?f=%5Cfrac%20%7B%282%5Cmu_s%5D%7D%7B%5B1-%5Cmu_s%5E%7B2%7D%5D%7D%20%3D%5Cfrac%20%7Bv%5E%7B2%7D%7D%7Brg%7D)
Therefore, the minimum radius of the curvature of the curve is
= ![\frac {v^{2}[1-\mu_s^{2}]}{2\mu_s g}](https://tex.z-dn.net/?f=%5Cfrac%20%7Bv%5E%7B2%7D%5B1-%5Cmu_s%5E%7B2%7D%5D%7D%7B2%5Cmu_s%20g%7D)
= ![\frac {(13.89 m/s)^{2}[1-(0.06)^{2}]}{(2)(0.06)(9.81)}](https://tex.z-dn.net/?f=%5Cfrac%20%7B%2813.89%20m%2Fs%29%5E%7B2%7D%5B1-%280.06%29%5E%7B2%7D%5D%7D%7B%282%29%280.06%29%289.81%29%7D)
= 163.3 m