Question

Find the maximum speed with which a car can turn on a bend without skidding if the radius of the bend is $20m$ and the coefficient of friction between the road and the tires is $0.4$.

Answer 1

Step 1: Given data

Radius of the bend, $r=20m$.

Coefficient of friction, $\mu =0.4$.

Step 2: Formula used

Centrifugal force,

$F_c=\frac{m{v}^2}{r}$

where $m$ is mass $v$ is velocity and $r$ is the radius of the bend.

Max frictional force,

$f_{max}=\mu N=\mu mg$

where $N$ is the normal force, $\mu$ is the coefficient of friction and $g$ is acceleration due to gravity.

Step 3: Calculation

As shown in the figure, centrifugal force $F_c$ acts on the car towards outside of the bend and frictional force $f$ acts towards the center. Here,

$$\begin{gathered} F_c=\frac{m{v}^2}{r} \\ {f}_{max}=\mu N=\mu mg \end{gathered}$$

Since they act opposite,

$$\begin{gathered} f_{max}=F_c \\ \Rightarrow \mu mg=\frac{m{v}^2}{r} \\ \Rightarrow v=\sqrt{\mu rg} \\ \Rightarrow v=\sqrt{0.4\times 20\times 10}\left(\because g=10m{s}^{-2}\right) \\ \Rightarrow v=\sqrt{80} \\ \Rightarrow v=4\sqrt{5}m{s}^{-1} \end{gathered}$$

Maximum speed with which the car can turn on the bend, $4\sqrt{5}m{s}^{-1}$.