Question

A cyclist speeding at $18km/h$ on a level road takes a sharp circular turn of radius $3m$ without reducing the speed and without bending towards the centre of the circular path. The coefficient of static friction between the tyres and the road is $0.1$. Will the cyclist slip while taking the turn?

• A. Yes
• B. No
• C. Cannot be judged
• D. None of these

Answer 1

The correct option is A Yes

$\text{Step 1: Identifying forces and Drawing free body diagram}$

$\text{Step 2: Applying Newton's law along Centripetal Direction}$

While taking sharp turn, The only possible force towards center is friction, Hence friction will provide centripetal acceleration.

Applying Newton's second Law on car along centripetal direction (considering direction towards centre as positive).

$\sum F_C=m{a}_c$

$\Rightarrow$ $f=m\frac{V^2}{r}$ $....\left(1\right)$

$\text{Step 3: Condition for Maximum Velocity without slipping}$

$\text{Back View:}$

For Maximum possible velocity, Maximum Static friction will act, Therefore $f=\mu N$

$\therefore$ $f=\mu mg$ $....\left(2\right)$ ($N=mg$ by applying Newton's Law in Vertical Direction)

From Eqn $\left(1\right)$ and $\left(2\right)$

$\mu mg=\frac{m{V}^2}{R}$

$\Rightarrow V_{max}=\sqrt{\mu Rg}$

$\text{Therefore, for no slipping:}$

$V\le V_{max}=\sqrt{\mu Rg}$

$\text{Step 4: Solving equation}$

$\mu =0.1$ $;$ $R=3m$

$V_{max}=\sqrt{0.1\times 3\times 10}m/s$

$=1.73m/s$

But

$V=18km/h=18\times \frac{5}{18}m/s=5m/s$

Since, $V>V_{max}$

Hence, The cyclist will Slip.

$\text{Alternate solution:}$

As the diagram(FBD) shows, if centripetal force $F_{net}$ applying on cycle is more than static friction $f$ then the cyclist will slip down while taking turn

$N=mg...\left(1\right)f=\mu N...\left(2\right)$

From above two equations

$\Rightarrow f=\mu mg=0.1\times m\times 10=m$

As per Newton's Second law (Solving in frame of reference of ground)

$\sum F=ma$

$\Rightarrow F_{net}=m\frac{v^2}{R}=m\frac{5^2}{3}=8.33\times m$

We can see that $F_{net}>f$ so the cyclist will slip down