Problem 197
What is the highest speed at which a car can drive around a…
What is the highest speed at which a car can drive around a horizontal circular path of radius $R = 30\ \mathrm{m}$ without skidding, if the coefficient of friction between the tyres and the road is $\mu = 0.25$?
Explanation of Method
Highest speed on a flat curve.
Step 1: Friction supplies the centripetal force
\[ \frac{mv^2}{R} \le \mu mg, \]
\[ v \le \sqrt{\mu g R} = \sqrt{0.25 \cdot 9.81 \cdot 30} = \sqrt{73.6}. \]
\[v_{\max} \approx 8.58\ \mathrm{m/s} \approx 30.9\ \mathrm{km/h}\]
$f = \frac{mv^2}{R}$
$v$
$R = 30\ \mathrm{m}$
$f \le \mu mg$
$v_{\max} = \sqrt{\mu g R} \approx 8.6\ \mathrm{m/s}$
Result
\[\boxed{\,v_{\max} \approx 8.58\ \mathrm{m/s} \approx 30.9\ \mathrm{km/h}\,}\]
Remark
Check. The original result is correct. $\mu = 0.25$ corresponds to a slippery (wet or icy) road; on dry asphalt ($\mu \approx 0.7$) the same curve could be taken at about 52 km/h.