Car on a flat curve

Problem

A car ($1200\;\text{kg}$) goes round a flat curve of radius $4\;\text{m}$ at $v = 6{,}9\;\text{m/s}$. The coefficient of friction is $\mu = 0{,}5$. Does the car stay on the road?

On the left, the top view (the centripetal acceleration points towards the centre O); on the right, the side view with weight $\vec{P}$, reaction $\vec{R}$ and friction $\vec{F}_a$ acting as the centripetal force.

Solution

Data. $m = 1200\;\text{kg}$, $r = 4\;\text{m}$, $v = 6{,}9\;\text{m/s}$, $\mu = 0{,}5$, $g = 9{,}81\;\text{m/s}^2$.

Centripetal acceleration required to maintain the circular path: $\abs{a_c} = \frac{v^2}{r} = \frac{6{,}9^2}{4} = \frac{47{,}61}{4} = 11{,}9\;\text{m/s}^2$

Centripetal force required: $F_c = m\,a_c = 1200 \cdot 11{,}9 = 14\,280\;\text{N}$

Force available. On a flat road the only force directed towards the centre is static friction. Its maximum value is (the normal reaction balances the weight, $R = mg$): $F_{a,\max} = \mu\,R = \mu\,mg = 0{,}5 \cdot 1200 \cdot 9{,}81 = 5886\;\text{N}$

Comparison. Since $F_c = 14\,280\;\text{N} > F_{a,\max} = 5886\;\text{N}$ friction is not enough to provide the required centripetal force.

Result. $\ev{F_c > F_{a,\max} \;\Rightarrow\; \text{friction is not enough: the car leaves the road.}}$

The maximum safe speed would be $v_{\max} = \sqrt{\mu g r} = \sqrt{0{,}5 \cdot 9{,}81 \cdot 4} \approx 4{,}4\;\text{m/s}$, well below the $6{,}9\;\text{m/s}$ held.

Links

Topics: Dinamica · Attrito Concepts: Moto circolare uniforme · Accelerazione centripeta · Forza centripeta · Attrito statico · Forza normale Skills: Applicazione delle leggi di Newton Objects: Curva sopraelevata