Practice FBD for circular motion with friction, find maximum safe speed along horizontal circular rough road and on banked road for given coefficient of friction.
\(N=\dfrac{mv^2}{R}\)
\(f_s=mg\)
\(N=\dfrac{mv^2}{R}\)
\(f_s=mg\)
\(f_s=\mu_sN\)
where \(\mu_s\) is the coefficient of static friction.
Putting value of \(f_s\) and N in \(f_s=\mu_sN\)
\(mg=\mu_s \dfrac{mv^2}{R}\)
\(v=\sqrt{\dfrac{Rg}{\mu_s}}\)
\(Nsin\,\theta+fcos\,\theta=\dfrac{mv^2}{R}\)
where v is the velocity of vehicle
R is the radius of curvature
\(Ncos\,\theta-fsin\,\theta-mg=0\)
Dividing (1) by (2)
\(\dfrac{sin\,\theta+\mu_s\;cos\,\theta}{cos\,\theta - \mu_s\;sin\,\theta} =\dfrac{v^2}{Rg}\) ... (3)
\(\Rightarrow sin\,\theta+\mu_s\; cos\,\theta = \dfrac{v^2}{Rg}cos\,\theta-\mu_s \dfrac{v^2}{Rg}sin\,\theta\)
\(\Rightarrow \mu_s = \dfrac{\dfrac{v^2}{Rg}cos\,\theta -sin\,\theta}{cos\,\theta+ \dfrac{v^2}{Rg}sin\,\theta}\)
\(\Rightarrow \mu_s= \dfrac{v^2cos\,\theta-Rg\;sin\,\theta}{v^2 sin\,\theta+Rg\;cos\,\theta}\)
\(Nsin\,\theta+fcos\,\theta=\dfrac{mv^2}{R}\)
where v is the velocity of vehicle
R is the radius of curvature
\(Ncos\,\theta-fsin\,\theta-mg=0\)
\(f_s=\mu_sN\)
\(Nsin\,\theta+\mu_sN cos\,\theta = \dfrac{mv^2}{R}\) ... (1)
\(Ncos\,\theta-\mu_sN sin\,\theta = mg\) ... (2)
Dividing (1) by (2)
\(\dfrac{sin\,\theta+\mu_s\;cos\,\theta}{cos\,\theta - \mu_s\;sin\,\theta} =\dfrac{v^2}{Rg}\) ... (3)
\(v^2=\dfrac{Rg(sin\,\theta+\mu_s cos\,\theta)}{(cos\,\theta-\mu_s sin\,\theta)}\)
\(v_{max}=\sqrt{\dfrac{Rg(sin\,\theta+\mu_s cos\,\theta)}{(cos\, \theta-\mu_s sin\,\theta)}}\)
A 12 m/s
B 14 m/s
C 16 m/s
D 10 m/s
\(f_s=\dfrac{mv^2}{R}\); \(N=mg\)
\(f_s\le \mu_smg\)
\(\dfrac{mv^2}{R} \le \mu_s mg\)
\(\mu_s \ge\dfrac{v^2}{Rg}\)
\(v^2 \le \mu_s Rg\)
Maximum safe speed
\(v = \sqrt{\mu_s Rg}\)
A \(10\ m/s\)
B \(15\ m/s\)
C \(20\ m/s\)
D \(30\ m/s\)