A track consists of two circular parts `ABC` and `CDE` of equal radius `R` and joined smoothly as shown. Each part subtends an angle `2theta` at center. A vehicle of mass `m` travels with constant force by the road on the vehicle when it is at (i) `B` (ii) `D` (iii) just before `C` and (iv) just after `C`. (b) find the friction by the track on the vehicle when it is at `B`, `C` and `D`. (c) what should be the minimum friction coefficient between the road and the vehicle, which will ensure that cyclist can move with constant speed.

Explanation

(a) At B: `mg-N_(B)=(mv^(2))/(R )impliesN_(B)=mg-(mv^(2))/(R )`
At D: `N_(D)-mg=(mv^(2))/(R )impliesN_(D)=mg+(mv^(2))/(R )`
just before C: `mg cos theta-N_(c)=(mv^(2))/(R )implies N_(c)=mg cos theta -(mv^(2))/(R )`
Just after C: `N'_(c)-mg cos theta=(mv^(2))/(R )`
implies `N'_(c)=mg cos theta+(mv^(2))/(R )`
(b) since the vehicle is moving with constant speed, i.e., the tangential acceleration is zero, i.e., net tangential force on it always is zero.
At `B` and `D`, no component of weight along the surface of track, hence friction is not developed.
At `C`, component of weight `mgsintheta` is balancedby friction so that tangential force is zero, `f=mg sin theta`.
(c ) Just before `C,f=mu_(1)N_(c)`, find `mu_(1)`.
just after `C,f=mu_(2)N'_(c)`, find `mu_(2)`.
since `N_(c)ltN'_(c)mugtmu_(2)`.
Satisfy the requirement of `mu_(1)` as well as `mu_(2)`, say `mu_(1)=0.80,mu_(2)=0.60`, minimum value of friction coefficient will be `0.80`.