Maximum speed on a flat curve

Problem

A car travels round a flat curve of radius $R = 40\;\text{m}$. The coefficient of static friction between tyres and road is $\mu_s = 0{,}50$. Determine the maximum speed at which the car can take the curve without sliding outwards. (Use $g = 9{,}8\;\text{m/s}^2$.)

Solution

Data. $R = 40\;\text{m}$, $\mu_s = 0{,}50$, $g = 9{,}8\;\text{m/s}^2$.

Set-up. On a flat road the only force directed towards the centre is static friction. The limiting condition occurs when friction reaches its maximum value and exactly provides the required centripetal force: $F_{a,\max} = F_c \quad\Rightarrow\quad \mu_s\,mg = \frac{m\,v_{\max}^2}{R}$

The mass cancels (the maximum speed does not depend on the mass of the car): $v_{\max}^2 = \mu_s\,g\,R \quad\Rightarrow\quad v_{\max} = \sqrt{\mu_s\,g\,R}$

Numerical substitution: $v_{\max} = \sqrt{0{,}50 \cdot 9{,}8 \cdot 40} = \sqrt{196} = 14\;\text{m/s}$ (about $50\;\text{km/h}$).

Result. $\ev{v_{\max} = \sqrt{\mu_s\,g\,R} \approx 14\;\text{m/s}}$

Links

Topics: Dynamics · Friction Concepts: Uniform circular motion · Centripetal force · Static friction · Normal force Skills: Symbolic set-up